The Reds were far from their best as Jurgen Klopp rotated his side with one eye on Wednesday's Champions League trip to Inter Milan.
But Klopp's men did just enough to keep City in their sights at the top of the table with a game in hand to come.
On the other end, Burnley remain bottom and are now seven points from safety after wins this weekend for Everton and Newcastle.
A week on from facing off in the final of the Africa Cup of Nations, Mohamed Salah and Sadio Mane were reunited on the same side by Klopp.
Diogo Jota, who scored both Liverpool's goals in a 2-0 win over Leicester in midweek, was surprisingly left out along with new signing Luis Diaz.
Burnley have still only won one league game all season, but had held Arsenal and Manchester United in their past two games and would have been worthy of a point once more with better finishing.
New signing Wout Weghorst was a constant menace until the Dutch international hobbled off with an injury that will worry Sean Dyche 15 minutes from time.
Weghorst should have done better when he skewed an attempted chip over Alisson Becker off target early on.
The Brazilian international was the busier of the two goalkeepers in the first half as he was also forced into action by Josh Brownhill and rushed from his goal to prevent Jay Rodriguez from rushing onto a long ball.
Liverpool had barely threatened the Burnley goal until they went in front five minutes from halftime.
Mane flicked on Trent Alexander-Arnold's corner and Fabinho swept in his fifth goal in seven games from close range.
Burnley continued to probe after the break, but their lack of goal threat was exposed.
Dyche's men have scored just 17 goals in 21 league games this season and are fast running out of time to extend their six-season stay in the top-flight.
Liverpool should have added a second late on when Salah set up Jota, only for his effort to be deflected behind by James Tarkowski.
But one goal was enough to keep alive their slim hopes of denying City a fourth Premier League title in five