The defender fired home in each half to score his first goals for the club and give Mark Hughes' men their first victory at the Stadium of Light on Tuesday after Jozy Altidore had put Sunderland ahead.
Sunderland reached the final last year before losing to Manchester City, and they appeared set to continue their recent love affair with the competition when United States international Altidore broke the deadlock in the 16th minute.
Altidore's goal marked his first for Sunderland since December last year, but it came in a losing cause as Muniesa turned the game in Stoke's favour.
The 22-year-old found his scoring touch on Wearside to keep alive Stoke's hopes of winning a cup that provided their sole major trophy success back in 1972.
Sunderland, for whom defender Sebastian Coates made his debut as one of seven changes from the weekend’s Premier League draw with Burnley, made the brighter start and went close to opening the scoring in the first minute when Billy Jones' deep cross just evaded Altidore at the far post.
Jack Rodwell then fired over the crossbar as the hosts pushed for the opener, which finally arrived in the 16th minute through Altidore. The American produced a smart turn and finish into the bottom corner to end his long goal drought.
But instead of pressing home their advantage, Sunderland instead conceded the initiative as Stoke were rewarded for a good spell of pressure by Muniesa after half-an-hour.
The Spaniard collected a lay-off from Oussama Assaidi and surged into the box before unleashing a fierce low shot that went under Costel Pantilimon's legs and into the net.
Muniesa's leveller opened up the game further, with both teams seeing chances go begging in the immediate aftermath.
First, Adam Johnson had a curling free-kick turned away by Jack Butland while, at the other end, Assaidi shot wastefully wide following a quick Stoke counter-attack.
Patrick van Aanholt was brought on for Coates at the break, but chances proved few and far between in a second period that was played at a considerably slower tempo than the first.
Stoke would have fallen behind again had it not been for a wonderful diving save from Butland to keep out Will Buckley's close-range effort, with the 24-year-old then denied by the England shot-stopper for a second time as Sunderland began to reassume control.
But their failure to beat Butland was duly punished 71 minutes in when Muniesa thumped home an emphatic left-footed finish after the referee had allowed play to go on, despite the former Barcelona man appearing to have fouled Johnson in the build up.
Marko Arnautovic then spurned a glaring chance, missing the target from point-blank range, but it did not matter as the visitors held on to go through.