The Reds made two changes as Roberto Firmino and Adam Lallana came in from the side that drew 0-0 in last weekend's Merseyside derby, while it was the same again for the visitors despite losing 3-1 at Crystal Palace last time out.
After four draws from six games to fall four points behind Manchester City, the hosts needed to win their game in hand and made the worst possible start as Ashley Westwood looped a corner kick over Alisson in the sixth minute.
That woke the home side up and they equalised after 19 minutes, Lallana's smart switch of play prompting nice work by Mohamed Salah and after Tom Heaton and James Tarkowski both failed to clear, Firmino could not miss.
Burnley looked dangerous on the break but failed to create chances and were punished on 29 minutes. A frantic piece of play resulted in neither Westwood nor Jeff Hendrick clearing and after a potential penalty shout was waved away, Sadio Mane seized on the loose ball and curled home.
That was enough for a 2-1 lead at the break but the second half struggled to find any real momentum for either side and it was almost a surprise when Firmino made it 3-1 on 67.
Inevitably it came from a mistake, this time Heaton scuffing a goal kick and while it could have been a foul on Salah by Charlie Taylor, the Brazilian took advantage and fired home.
The highlight after that was former Red Peter Crouch being introduced to rapturous applause by the Anfield faithful and the game appeared to be crawling to its death.
However, there was almost a late twist when Johann Berg Gudmundsson made it 3-2 in stoppage time but Mane grabbed his second soon after to ensure Liverpool closed the gap at the top of the table back to a single point behind City.