Diogo Jota came off the bench to score on his debut as Liverpool recovered from a goal down to beat Arsenal 3-1 at Anfield.
French striker Alexandre Lacazette opened the scoring against the run of play after an error from Robertson.
But the champions responded with Mane's tap-in and Robertson redeemed himself with a neat finish before the break.
Jota struck late on to ensure Liverpool took home all three points and maintained their perfect start to their title defence.
Liverpool received a huge boost when goalkeeper Alisson Becker was passed fit to start in goal, but Thiago Alcantara missed out on the 18-man squad due to a lack of sharpness.
David Luiz and Kieran Tierney returned to the Arsenal backline, while Mohamed Elneny came in for Dani Ceballos.
Having gone 60 league games and three-and-a-half years unbeaten at Anfield, it seemed like Mikel Arteta's side could provide a stern test to their invincible run. After all, they had won both of their opening matches in contrasting styles, pointing to a new aura to this Arsenal team.
But as we have often seen from the Premier League champions, their ability to press and hound players into making mistakes is unrivalled and Arsenal faced a difficult opening quarter of an hour stranded on the edge of their own box.
The first real chance of the game arrived on 12 minutes when Trent Alexander-Arnold's deep cross found Virgil van Dijk at the far post and his header was cleared by David Luiz.
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Klopp's side continued to build their attacks again and again, with Alexander-Arnold's drilled effort taking a wicked deflection off Hector Bellerin only to bounce back off the crossbar.
With Liverpool enjoying almost total domination, it felt like only a matter of time before the opener would arrive. But to Arsenal's delight, it came against the run of play.
Worked out from their own corner flag, Arsenal found a way past Liverpool's high press and Ainsley Maitland-Niles tried to pick out Willian at the far post, but Robertson got in the way. The Scot botched his clearance, however, and there was Lacazette to scuff his volley past Alisson.
It seemed impossible to think Liverpool wouldn't get a chance to level having been on top and as it often goes at Anfield, the lead lasted just a few minutes. Mohamed Salah got the beating of Kieran Tierney down the right, fired at Bernd Leno and the German stopper's save landed at the feet of Mane, who couldn't miss from a yard out.
The instant response revitalised the hosts and no sooner had Arsenal regrouped, Andy Robertson took advantage of some slack marking from Hector Bellerin to dink over Leno to turn the game on its head.
After spending such a long period of the game on the back foot, Arsenal would have been grateful for still being in the game at 2-1 and Arteta sent his side out with a more positive mentality.
Around the hour mark, a flurry of a chances arrived. First, Van Dijk hit one from range and then Lacazette, springing the offside trap, couldn't find a way past Alisson Becker.
With his side showing encouraging signs since the break, Arteta sent on Nicolas Pepe and Dani Ceballos in search of a leveller, and Ceballos nearly made an instant impact, again sending Lacazette through on goal but the Frenchman inexplicably couldn't beat Alisson.
Diogo Jota made a late cameo for his Liverpool debut and almost marked the occasion with a goal, only to strike into the side netting. But he would get his moment two minutes from time, volleying home into the bottom corner to seal a deserved victory for Klopp's men.
Stick with Standard Sport for all the latest reaction.
Liverpool team news
Starting XI: Alisson, Alexander-Arnold, Gomez, van Dijk, Robertson, Keita, Fabinho, Wijnaldum, Salah, Firmino, Mane
Subs: Adrian, Milner, Jones, Minamino, Jota, Origi, Neco Williams
Klopp bullish over title hopes
Regarding Liverpool's title chances, Klopp said: "I don't know why we speak about winning the league after two matchdays.
"A lot of times it is an open race. Chelsea will be there 100 per cent, (Manchester) United, Arsenal, Tottenham - these teams will always be there.
"Then there are tough moments for us and others in a season and it is about how you react. The season is not a sprint it is a marathon and you have to go through a lot of different moments.
"I don't know who will win the league at the end of the season but I know a few teams will try and we will be one of them."
Arteta wants substance and style
Mikel Arteta insists he is happy to take a pragmatic approach if it brings success to the Emirates Stadium.
"Sometimes it's what you want to do as a coach and sometimes it's what you are allowed to do with the levels of players and performance that top teams can do against you," he replied when asked about moving away from the previous approach.
"But we have to prepare the games, finding scenarios to win. I'm here to win. I'm here to find ways to do it. If I am proposing something very difficult that is exposing our team right now, I don't think I'm acting in the best favour of the club or the players.
"The players we have available can change our game plan. We have to be able to adapt. I think in football now one way doesn't work unless you are superior to the opponent all the time."
Anfield fortress
Jurgen Klopp insists there is no "secret recipe" to Liverpool's incredible unbeaten home form.
The Reds are closing in on the club record of 63 home league wins, set between 1978-81 and the last time they were beaten at Anfield was when they lost 2-1 to Crystal Palace in April 2017.
"The only time I think about that number is when someone asks me about it and once I've answered the question I forget it immediately because I am absolutely not interested," he added.
"We did what we did. We try to win football games and we've done that at home in different circumstances, at nighttime, lunchtime, with supporters, without supporters.
"It is our home and we feel really well there but it is no secret recipe what we have - we never expect to lose but we don't expect to win.
"It is just we know we have to work and we want to deserve three points, that's what we want. That is it, pretty much."
We are expecting full lineups in around an hour.
We will have expert analysis from David Lynch (Liverpool) and Simon Collings (Arsenal) tonight, so watch this space...
How Arsenal could line up tonight...
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Arsenal team news
Mikel Arteta's transformation of Arsenal faces one of its biggest tests tonight as the Gunners visit Anfield looking to end a run of more than eight years without a win at Liverpool.
Arteta was in the team the last time the north Londoners left Anfield with three points, and their run against big six sides away from home has been a huge barrier to success in recent years.
The former Arsenal captain has, however, beaten Liverpool in his two games against the Reds as Gunners manager, with a Premier League win during Project Restart and a penalty shootout victory in the Community Shield last month.
Arsenal have stuck with a back three shape this season as Arteta looks to close the gap on Manchester City and Liverpool this season.
With David Luiz having returned from a neck problem in the midweek Carabao Cup win over Leicester, he will be hoping to feature on Monday, while Arteta will be leaning on the experience of Willian to break an unwelcome record.
How Liverpool could line up tonight...
Here's how we have the hosts should Alisson and Thiago not be deemed fit enough to start.
Liverpool team news
Liverpool will be without captain Jordan Henderson for tonight's meeting with Arsenal.
The midfielder was forced off at half-time of the Reds’ win at Chelsea last weekend after feeling tightness in his thigh, with Thiago Alcantara making his bow from the bench as a result.
And Klopp has revealed that Henderson’s recovery has not moved quickly enough for him to be involved when the Gunners visit Merseyside early next week.
The same goes for Joel Matip, whose hamstring injury is likely to keep him out until after the next international break.
But there was better news at the back, with both Joe Gomez and Billy Koumetio returning to training on Friday after a brief lay-off.
ith Gomez only just back in training, Klopp may prefer to once again use Fabinho as a makeshift centre-back.
However, reports suggest Thiago could miss out on Monday along with first choice goalkeeper Alisson which could require Fabinho to shift back into midfield.
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Welcome to Standard Sport's LIVE coverage of Liverpool vs Arsenal.
Stick around, we have all your team news and build-up throughout the day.