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Mohamed Salah, who scored twice, grabbed the 69th-minute winner for Liverpool after Salzburg, managed by American Jesse Marsch, had stunned Anfield with three goals either side of half-time to make it 3-3.
The league leaders, who won the Club World Cup title by beating Brazil's Flamengo last Saturday, take on second-placed Leicester City on Thursday and host Wolverhampton Wanderers on Sunday.
Chelsea's poor domestic form continued with a shock 1-0 home defeat by lowly Bournemouth, while Sheffield United moved up to fifth with a 2-0 win over fellow promoted side Aston Villa.
Even with strikers Mo Salah and Roberto Firmino warming the bench, Juergen Klopp's side had far too much firepower with Divock Origi scoring twice as Liverpool set a new club record of 32 league games undefeated.
Ten-man Manchester City gave up a two goal lead as they crashed to a 3-2 defeat by Wolverhampton Wanderers at Molineux on Friday
The victory takes Pep Guardiola's side, who had drawn at Newcastle United on Saturday, into provisional second spot and back within eight points of leaders Liverpool ahead of the Merseyside derby with Everton at Anfield on Wednesday.
Lallana says the team is not looking too far ahead and are fully focused on Thursday's league match against Sheffield United and Sunday's FA Cup tie against Everton.
Manchester City provided a timely reminder that they may still pose the greatest threat to Liverpool in the Premier League title race.
Juergen Klopp's side needed a point in Austria to be sure of qualification and after missing several good chances in the first half, they ended up with a comfortable win which should have been by a more convincing margin.
Crystal Palace claimed a first league win in six games with a 2-0 victory at Burnley and Southampton came from behind to beat fellow strugglers Watford 2-1.
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