The Serbia international netted from close range on nine minutes when Fabio Borini’s through pass deflected into his path as the Reds recorded their sixth win in nine visits to the Stadium of Light.
Liverpool completely dominated the first half, with Markovic denied a penalty in the early exchanges, but they wasted a plethora of chances that would have put the game beyond their hosts.
Sunderland were booed off at half-time after failing to muster a shot on goal and they were reduced to 10 men on 49 minutes when Liam Bridcutt picked up a second yellow card.
The sending off spurred Sunderland momentarily, with Adam Johnson crashing a shot off the crossbar, but that was as good as it got as their dismal home form continued.
Liverpool travelled to a blustery north east in good form and they completely dominated the first half.
The visitors should have been awarded a penalty after just three minutes when Markovic hit the deck following a clumsy challenge by Wes Brown, but referee Craig Pawson waved away their appeals.
But Markovic wouldn’t be denied and his regular runs up the right wing were rewarded when he fired Liverpool in front.
The Serbian passed inside to Steven Gerrard and then carried on forward looking for a return ball. Fabio Borini was taken out by John O’Shea on the edge of the area but Pawson played advantage and Markovic bundled past Costel Pantilimon.
The only negative from Liverpool’s first-half performance was that they didn’t extend their lead. Markovic hit the crossbar with a spectacular effort from the edge of area, while Gerrard twice forced Pantilimon into action and also smashed a shot narrowly wide.
Liverpool should have doubled their lead on the stroke of half time when former Sunderland loanee Borini beat Brown to Emre Can’s through ball. The Italian rounded Pantilimon but fired into the side netting.
Gus Poyet would have been glad to get his team into the dressing room but his half-time teamtalk went up in smoke on 49 minutes when Bridcutt picked up a second yellow card for a needless foul on Can.
Despite their numerical disadvantage, Sunderland then enjoyed a spell of possession and Johnson went close to a stunning equaliser when his long-range shot cannoned back off Simon Mignolet’s crossbar.
Liverpool continued to create chances, with Coutinho shooting narrowly wide and Pantilimon denying substitute Mario Balotelli from an acute angle, but their failure to register a second goal kept Sunderland in the game.
Sunderland staged a last-gasp assault with Pantilimon going forward for a corner, but Liverpool clung on for a win that should have been much more convincing.
Sunderland: Pantilimon (8), Vergini (6), O’Shea (6), Brown (7), Van Aanholt (6), Johnson (7), Bridcutt (4), Gomez (5), Larsson (6), Giaccherini (6), Wickham (5).
Subs used: Buckley (6), Graham (6), Mandron (6).
Liverpool: Mignolet (6), Can (7), Skrtel (7), Sakho (7), Markovic (8), Henderson (7), Lucas (6), Gerrard (7), Moreno (6), Coutinho (8), Borini (6).
Subs used: Lovren (6), Balotelli (6).
MAN OF THE MATCH: Lazar Markovic (Liverpool)