The 32-year-old volleyed home with two minutes left in the first half to give his side the points over the Blades.
Although the visitors fought back in the second period, they struggled to create clear opportunities without the threat of a confident striker such as Stewart.
Both sides have FA Cup fourth-round ties next week - after beating top-flight opposition in the previous round - but priorities for both is consistency in the Coca-Cola Championship.
The difference between the two promotion hopefuls was their attacking power. While Andy Gray toiled for the visitors as a lone striker, Stephen Elliott and Stewart were combining to great effect for Mick McCarthy's side.
Although Stewart was the only player to show composure in the opening exchanges, it was Elliott who had almost opened the scoring in the 15th minute.
Racing onto his strike-partner's intelligent flick following a clearance from a corner, the former Manchester City striker chipped Paddy Kenny with confidence but was denied his 13th goal of the season by the crossbar.
The front pair combining put the Black Cats in the ascendancy, and they had a great chance with a bizarre opportunity after 25 minutes.
Referee Peter Walton penalised Leigh Bromby for obstruction in the penalty area after Dean Whitehead's surging run. The resulting free-kick was taken inside the six-yard box and Kenny bravely stopped Arca's drive.
Neil Warnock's side are the league's away-day specialists, with eight wins on their travels this season, but they found opportunities difficult to forge.
Growing increasingly frustrated with his side's lack of enterprise, Alan Quinn was booked for a trip on Carl Robinson, and the home side continued to press. Michael Tongue had a weak effort easily saved by Thomas Myhre before the hosts took control of the game.
Stewart was sent through by Elliott after 36 minutes but dragged his shot wide with Kenny beaten. But the former Ipswich striker was not to be denied two minutes before half-time, crisply volleying home from Elliott's cross from the left, leaving Kenny with no chance.
Warnock responded by bringing on Danny Cadamarteri at the break for Andy Liddell - and the visitors looked to have more of an attacking threat. Three minutes into the second half Gray's turn and shot flashed wide with Myhre stranded.
But Sunderland did not try to defend their slender advantage, Stewart denied by a last-ditch Morgan tackle after Kenny had raced out of his area and misjudged a clearance.
Both sides were looking likely to grab the second goal of the game, with Gray unlucky to see his firm downward header from a corner blocked by team-mate Cadamarteri after 63 minutes.
McCarthy's tactic was to attack, bringing on Chris Brown for Jeff Whitley, who had been booked, in the 68th minute. Sean Thornton was brought on for Stephen Wright at the same time as the hosts looked to kill the game with a second goal.
Jonathan Forte was brought on for Quinn as the Blades looked for the leveller - and the youngster went on an exciting run on the left before mis-hitting his cross when in a good position.
Forte was then denied by Myhre after riding two challenges and firing a shot towards the top corner.
Warnock's last gamble was throwing on striker Paul Shaw for defender Danny Cullip - but it was Sunderland who came closest to scoring. Brown was found unmarked in the area by Dean Whitehead but Kenny produced a magnificent double save to keep it 1-0. Stewart was taken off to a standing ovation with five minutes remaining - his work was done.