Sadio Mane and Roberto Firmino's goals gave Liverpool a 2-1 win at Southampton despite a late scare from the hosts.
A Liverpool performance which had started so poorly on the south coast still held a half-time lead thanks to Mane's curling strike just before the break, and they marked a much-improved second half with a smart finish from Firmino (71).
But goalkeeper Adrian, who passed a late fitness test to start the game, had a moment to forget when his clearance rebounded in off Danny Ings to set up a tense ending, during which the ex-Liverpool forward missed a golden chance to level from close-range.
Southampton: Gunn (7), Valery (7), Bednarek (5), Vestergaard (6), Yoshida (5), Betrand (6), Hojbjerg (6), Ward-Prowse (6), Romeu (7), Redmond (7), Adams (5).
Subs: Ings (6), Armstrong (5), Djenepo (6).
Liverpool: Adrian (6), Alexander-Arnold (6), Matip (5), Van Dijk (7), Robertson (7), Milner (6), Wijnaldum (6), Oxlade-Chamberlain (7), Salah (7), Firmino (7), Mane (8).
Subs: Fabinho (6), Origi (6), Henderson (n/a).
Man of the match: Sadio Mane
Southampton were left with nothing to show for a spirited effort come the full-time whistle, while Liverpool continued their perfect start and avoided any major issues from their 4,000-mile round trip to Istanbul for the Super Cup last Wednesday.
Liverpool ride luck to stay perfect
Whether a result of their excursion to eastern Europe in midweek or not, Liverpool were well below their best and struggled to get any of their potent front-line into the game.
Their cause was not aided by James Milner's lengthy absence to receive stitches in a head injury, and with a man extra Southampton came close to an opener when James Ward-Prowse's corner was met by Maya Yoshida, but an instinctive save from Adrian kept the ball out.
Che Adams later should have opened his Saints account with another header from Oriel Romeu's cross but nodded over the bar unmarked from 12 yards.
With Liverpool unpunished for their sluggish start, they struck a cruel blow on the stroke of half-time. Mane was given space on the edge of the area from a throw-in and curled beyond Angus Gunn to dramatically alter the course of the game.
The half-time break did nothing to stop Liverpool taking full advantage of the momentum a goal can give, and suddenly they looked the side in full control.
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It looked like the old Liverpool when Mane's clip over the top allowed Mohamed Salah to run through on goal before Gunn saved with his legs, but the front three didn't need to be at their best to double their lead. Mane robbed Jan Bednarek from a Southampton throw and cut in from the left before firing across the goalkeeper.
Firmino and Andrew Robertson missed chances to add to the scoreline with Southampton looking deflated, but they had reason to believe again when Adrian's howler of a clearance struck Ings and rebounded in.
Ings was a whisker away from stunning his former club minutes later when Jan Valery's low cross found him, but he side-footed wide from a great position.
Ralph Hasenhuttl: "If you play against a big team, you need a good performance to take something. We had a fantastic first half, a lot of chances and stressed them very early, invested a lot. A lot more than against Burnley, and I had a good feeling.
"But Mane scored a fantastic goal, these strikers are unbelievable, they do not need time and space and the second half was more difficult when they became more comfortable."
Jurgen Klopp: "It looked like a proper, proper banana skin since Wednesday night, it was about headlines and they had already been prepared. My favourite headline is the mentality giants are in town, and that's the game I saw.
"I'm really proud of what the boys did, I was part of the travel and what the boys did. We deserved the three points, no doubt about that, they had a little rise when Adrian helped them with the goal, but he's now arrived at Liverpool."
- Liverpool have now won 11 consecutive Premier League matches, equalling their best -ever winning streak in the competition, previously set between February and April 2014 under Brendan Rodgers.
- Southampton have lost both of their first two Premier League games of a season for the first time since their first season back in the division in 2012-13 under Nigel Adkins.
- Sadio Mané scored his 20th goal for Liverpool in all competitions in 2019, at least eight more than any of his teammates in this calendar year (Mo Salah 12).
Man of the match - Sadio Mane
The man who changed the game, Mane's goal gave the match a very different complexion at the break and was a touch of class which Liverpool had lacked in the opening 45 minutes.
He added another major contribution after the break by hassling Bednarek into a mistake which led to Firmino's goal and was a general nuisance in his first Premier League start of the season. 20 goals from his last 26 games is a formidable return.
Liverpool host Arsenal live on Sky Sports in the tea-time kick-off next Saturday at 5.30pm. Earlier that day Southampton travel to Brighton at 3pm.