Danny Ings hit the post from distance, with Ashley Barnes firing the rebound over as the Clarets dominated.
Scott Arfield went close, before Sterling rounded Tom Heaton following Philippe Coutinho's clever flick.
Liverpool keeper Brad Jones was forced off with a thigh injury and substitute Simon Mignolet was far from convincing.
Liverpool climb to ninth in the Premier League but it was an uncomfortable afternoon for Brendan Rodgers's side, who were poor for long periods.
Burnley, who drop to 19th, were dominant and showed great heart and desire.
But their inability to take chances cost them dear and they face starting 2015 in the relegation zone unless they can pull off one of the surprises of the season at Manchester City on Sunday.
Liverpool might have finished on a high in their previous outing against Arsenal but they were fortunate not to fall behind early in east Lancashire.
Ings hit the woodwork with a low 25-yard drive after spinning into space with Jones rooted to his line, Barnes blazing the rebound over from close range.
Moments later Jones was forced off with a suspected thigh problem, paving the way for Mignolet's return to action for the first time since 9 December.
Yet the Belgian was fortunate to escape unpunished as Ings charged down a kick by the substitute keeper before the hosts carved out a succession of chances.
Liverpool were relieved to be on level terms at the interval, with Burnley dominant in every aspect.
But the Clarets have struggled for goals this season.
First Barnes fired narrowly over the bar, before Arfield's shot deflected off Lucas and whistled narrowly wide. George Boyd then went close from 20-yards.
Yet Liverpool secured the points when Coutinho lofted a ball into Sterling's path, the forward rounding the keeper to score.
Still Burnley threatened.
Mignolet took too long to clear the ball, with Michael Keane heading a good chance into the side netting from the resulting corner.
Forward Sam Vokes, who suffered cruciate knee ligament damage in March, made his first appearance of the season as a substitute.
But Burnley were unable to force an equaliser as Liverpool secured only their fifth league clean sheet of the season.