The England forward rounded Tom Heaton before finding the net after Danny Ings had earlier fired a low shot against the upright, while Scott Arfield and George Boyd went close in a game largely dominated by the hosts.
Liverpool were under pressure for most of the match and will feel lucky to have held on to pick up a victory ??? their first in four Premier League games.
The hosts, meanwhile, remain in the relegation zone with defeat but will be encouraged by a very positive display.
Both sides named unchanged line-ups, with Brendan Rodgers keeping faith with the 3-4-3 formation that saw them earn a last-ditch point against Arsenal. Burnley, meanwhile, welcomed Sam Vokes back into the squad after a nine-month absence with a cruciate ligament injury.
Burnley started the far brighter of the two teams, and Ings almost gave the hosts a deserved lead as he received a throw before spinning on the spot and firing a low effort which hammered back off the near post.
Brad Jones was nowhere near the striker???s effort, and it soon became apparent the goalkeeper was not completely comfortable, coming off with a thigh injury moments later to be replaced by the much-scrutinised Simon Mignolet.
The hosts continued to press, and Ashley Barnes headed over after setting a sweeping move going, finding Dean Marney out wide before meeting the return ball to put it over the crossbar.
Dyche???s men went close once more 10 minutes before the break as they won a corner, and Arfield met a Liverpool clearance to volley a low effort which just deflected wide off the outstretched foot of Lucas.
Both Dean Marney and George Boyd fired efforts over as Burnley tried to make their possession and territory count, but they couldn???t find a way through before the break.
Their pressure continued, albeit with less intensity, in the early stages of the second half, but the sucker punch came on 63 minutes as Liverpool moved in front.
Philippe Coutinho flicked on Lucas??? header forward, and Sterling raced on to the ball beyond the defence, rounding Tom Heaton before firing the ball low into the empty net to give the visitors a surprise lead.
The hosts looked dejected and struggled to fashion another decent chance to level matters, as Liverpool started to keep the ball with more authority.
Coutinho fired a deflected effort wide inside the final 20 minutes, before substitute Rickie Lambert had the ball in the back of the net with a wonderful first-time shot, but the striker was denied a goal as the flag went up.
Burnley continued to press as Vokes was given his first appearance in nine months, but Dyche’s men ran out of steam as Liverpool held firm during the latter stages and gave Rodgers a much-needed victory.