Markovic magic enough for Liverpool in scrappy affair at the Stadium of Light
The £20million summer signing from Benfica fired home in the ninth minute but despite a dominant performance the visitors missed numerous chances to make the game safe against the below-par hosts.
And after Liam Bridcutt was harshly red-carded in the 49th minute for tripping Emre Can it was Gus Poyet's men who came closest to a leveller when Adam Johnson clattered a long-range effort against the bar, but it was to remain 1-0 to the visitors.
Marshalled by the excellent Steven Gerrard, Liverpool were in almost total control in the first period and Sunderland's lacklustre series of defensive mistakes saw them booed off at the interval.
It took just three minutes for the visitors to fashion their first chance as Markovic had strong penalty claims turned down after falling under the challenge of Wes Brown in the box.
But five minutes later the Serbian produced a clinical finish to put his side ahead after Fabio Borini threaded a neat pass from the right side of the Sunderland box.
The way in which Markovic out-muscled a pair of home defenders in order to get his shot on target must have worried Poyet and there was worse to come.
After Larsson fired a long-range free-kick into Simon Mignolet's arms, the visitors took control of the game with Gerrard inevitably integral to the majority of their forward moves.
The Los Angeles-bound star shot just wide of Costel Pantilimon's left-hand post in the 24th minute and five minutes later saw a free-kick from the left edge of the box flapped over the bar by the 6ft 8ins keeper.
From the resulting corner, the lively Markovic came close to grabbing his second when his impressive scissors kick slammed against the crossbar from 25 yards.
Two more efforts from Gerrard underlined the visitors' dominance and former Sunderland loan star Fabio Borini came close two minutes before the break when he raced onto a long ball from Can and rounded Pantilimon but shot into the side-netting from a tight angle.
If Poyet hoped to use the break to ease the sluggishness out of his side he suffered a major blow just four minutes after the re-start when Bridcutt was dismissed for a second bookable offence after tripping Can on the Liverpool right.
With Gerrard substituted at the interval, Liverpool looked less effective going forward and they almost paid the price as Johnson lashed a left-foot shot from long-range which clattered the underside of Mignolet's bar before squirming to safety.
A brief promising spell having come to a close for the hosts, normal service looked set to resume when Mamadou Sakhoo strolled unchallenged over the half-way line and sparked a move which ended in Philippe Coutinho sweeping 66th minute effort wide when he should have scored.
Rodgers looked to liven up his forward line by replacing Borini with Mario Balotelli in the 68th minute and the Italian soon brought a save out of Pantilimon after wriggling through the Sunderland defence.
Despite their numerical disadvantage the hosts had were undoubtedly better in the second half and came close to snatching an equaliser in the 76th minute when captain John O'Shea swung and missed at a Larsson corner in a crowded box.
Sunderland staged a last-gasp assault with Pantilimon going forward for a corner but the visitors clung on for a win they ought to have made much more convincing.