Pep Guardiola's side eased to a 3-1 victory over Watford on Saturday to turn up the heat on Liverpool, who had drawn four of their previous six league matches.
The pressure was increased when Ashley Westwood scored straight from a corner on Sunday, but Roberto Firmino and Sadio Mane each netted twice to secure the comeback, despite a late Johann Gudmundsson goal amid difficult conditions on Merseyside.
Anfield was stunned after only six minutes when Westwood's corner from the left curled all the way into the net, although Alisson was furious it was allowed to stand after claiming he had been blocked by three Burnley players on the line.
Liverpool levelled within 13 minutes, though, Firmino tapping in after Tom Heaton made a mess of trying to block Mohamed Salah's low cross from the right.
10 - Roberto Firmino has now scored 10+ goals in all four of his Premier League seasons with @LFC, the best 100% such record of any player for the club in the competition. Consistency. #LIVBUR pic.twitter.com/zwmxkSr2DB— OptaJoe (@OptaJoe) March 10, 2019
And it was 2-1 before the half-hour mark, Mane side-footing into the right-hand corner after Adam Lallana did well to block a clearance.
Burnley improved after the break, with Dwight McNeil causing problems down the Liverpool right, but the result was put beyond them 67 minutes in when Firmino buried his second into an unguarded net after Salah had failed to capitalise on Heaton's poor goal kick.
Mane should have scored a late fourth when he somehow turned Trent Alexander-Arnold's cross onto the crossbar with the goal at his mercy, but he made amends by rounding Heaton and burying a left-foot finish after Gudmundsson's close-range finish had threatened to make the final few moments anxious.
What does it mean? Liverpool showing steel in title scrap
This was a real test for Jurgen Klopp's side, not least because of the wind and hail making life difficult even before Westwood's fortuitous opening goal.
They showed commendable resolve to force their way back into the match, though, and were good value for the three points, even if the result owed much to Heaton's errors.
They are back to within a point of City and 12 above third-place Tottenham with eight games to go, while Burnley are only two points above the bottom three.
Lallana stakes his claim
A few will have been surprised to see Lallana given only his fourth league start of the season, but he repaid Klopp's faith with an all-action display that should give him a chance of starting against Bayern Munich in midweek.
Heaton howlers leave England spot uncertain
Heaton's form had made him a candidate for the England number one spot in this month's international break, but he was directly to blame for two of Liverpool's goals here.
Key Opta facts
- Liverpool’s tally of 73 points from their opening 30 league games this season is their joint-best ever total at this stage of a league campaign (adjusting to three points for a win), level with the 1904-05 second tier season and 1987-88 in the top-flight.
- Liverpool are the first side to win four consecutive home Premier League games in which they conceded the first goal since Chelsea won six in a row between April 2013 and May 2015.
- Burnley have dropped more points from winning positions in their 30 Premier League games this season (10) than they did in the whole of 2017-18 (nine).
- Liverpool have now gone 36 Premier League games without a loss at Anfield (W26 D10), the joint-third longest unbeaten home run in the competition’s history, behind only Chelsea (86 games between March 2004-October 2008) and Manchester City (37 between December 2010-December 2012).
- Liverpool have conceded nine goals in their 10 Premier League games in 2019, one more than they had in their 20 league games this season before the turn of the year.
- Sadio Mane has become the fifth Liverpool player to score in six consecutive home appearances in the Premier League, after Michael Owen, Fernando Torres, Luis Suarez and Mohamed Salah.
- Sadio Mané has now scored 50 goals for Liverpool in all competitions in what was his 109th appearance for the club.