City, who had earlier beaten Everton 2-0, are two points behind Liverpool with a game in hand as they chase an unprecedented fourth consecutive English top-flight title.
The Reds were gifted the lead when Burnley goalkeeper James Trafford failed to claim the ball from a Trent Alexander-Arnold corner and Diogo Jota headed home his ninth Premier League goal of the season.
But they were pegged back just before half-time when Dara O’Shea powered a bullet header past Liverpool goalkeeper Caoimhin Kelleher, replacing Alisson Becker, who was missing with an illness.
Liverpool, playing in front of a record league crowd at Anfield of nearly 60,000 after a ground expansion, were a different proposition after the break and got their reward when Luis Diaz headed them back in front in the 52nd minute.
The visitors had several inviting chances to equalise before Darwin Nunez’s header from Harvey Elliott’s cross secured the vital win.
“It’s not only the three points,” Klopp told the BBC. “You cannot have momentum guaranteed all the time.
“Nine days ago we played incredible against Chelsea then we lost to Arsenal. It’s always a different challenge. If it’s not clicking straight away, fight through it.
“It was really tough today but the boys fought through and got three points.”
Haaland double
Earlier, Erling Haaland scored his first goals since November as Manchester City topped the table for the first time since November.
Pep Guardiola’s team laboured to break down the stubborn visitors at a nervous Etihad Stadium but Haaland struck twice late in the second half.
They are his first goals since returning from an absence of nearly two months due to a foot injury.
Guardiola’s men have won their past 10 games in all competitions but the Spaniard might be a little concerned with the way they laboured against struggling Everton, who remain in the relegation zone.
“In the first half our body language was not good,” said the City boss. “You cannot play good football if you are not positive. You do your best when you are positive. When you start to complain and are sad it does not work.
“We talked at half time and I told them, ‘what do you expect, to come here and win 5-0? Come on. Life is tough and football is tough’.”
Brennan Johnson struck a dramatic winner in the sixth minute of stoppage time to fire Tottenham back into the top four with a dramatic 2-1 win against Brighton.
The teams looked set to share the spoils after Pape Sarr cancelled out Pascal Gross’s early penalty in the 61st minute.
But South Korean star Son Heung-min, back from Asian Cup duty, teed up fellow substitute Johnson in the dying seconds to earn a vital victory.
“It’s fair to say that we are still a work in progress but we had a world-class player in Son, who makes the most difficult ball look simple to set up Brennan,” said Spurs boss Ange Postecoglou.
“Going into the game we looked at the squad and knew we had the ability to change it up. It’s fair to say that helped us.”
Newcastle beat Nottingham Forest 3-2 in the evening kick-off.
Brazil midfielder Bruno Guimaraes scored twice for Eddie Howe’s team, who climb to seventh in the table, but Forest have now taken just one point from their past four matches.
Bottom side Sheffield United beat Luton 3-1 and Rodrigo Muniz scored twice as Fulham beat Bournemouth by the same scoreline.
Brentford saw off Wolves 2-0 courtesy of Christian Norgaard’s first-half header and a fine team goal late in the game finished off by Ivan Toney.
Third-placed Arsenal travel to West Ham on Sunday while high-flying Aston Villa host Manchester United.
© Agence France-Presse
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