Sterling was on target early on, but the midfielder had been flagged for offside well before he got his shot away, ruling out the finish.
The home side had a decent chance early on when Danny Ings was left with time and space in the box, but the striker’s finish was kept out by the post.
Brendan Rodgers was forced to make an early change when Brad Jones aggravated a knock that he had picked up in the warmup, forcing the manager to bring on Simon Mignolet.
Liverpool started to dominate possession as the first half progressed, but Adam Lallana could not beat the keeper after being set up by a Steven Gerrard through-ball.
Scott Arfield came close just before half time, but the striker’s powerful finish was blocked by Lucas, who was back to help Mignolet.
Liverpool continued to see more of the ball at the start of the second half, but created few chances against Sean Dyche’s men.
The visitors eventually broke the deadlock though when Philippe Coutinho launched a high ball forward for Sterling, who rounded the keeper before tapping in.
Mignolet came under pressure as Burnley attempted to get back on level terms, but Kieran Trippier’s cross was headed into the side-netting by Michael Keane.
Rickie Lambert thought that he had put the win beyond doubt when he chipped in from a cross, but the substitute was a yard offside as the ball was played in.
Ings came close down the other end a few moments later with a rare second-half chance for the hosts, but he was unable to force a save from Mignolet.
The result moves Liverpool up to ninth in the Premier League table, while Burnley sit 19th.
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