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.
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.
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.
