Welbeck's fifth goal of the season saw Steve Bruce's Black Cats strengthen their claims on Europe.
The striker, on loan from Manchester United, stooped to head home from close range after Jussi Jaaskelainen had turned away Darren Bent's 32nd-minute shot.
However, keeper Craig Gordon had to pull off a superb 45th-minute save to deny defender Zat Knight an equaliser on the stroke of half-time.
Victory saw the Black Cats climb above the visitors into sixth place to cement their excellent start to the season as they gave a crowd of 35,101 something to cheer on a bitterly cold Wearside afternoon.
They might have won more comfortably had Welbeck's 70th-minute effort crept inside the post rather than coming off the outside of it, but they were good value for the points as they tamed Owen Coyle's high-flying side.
Bolton lost 4-0 at the Stadium of Light last season on an afternoon when Steve Bruce's men ended a 14-game wait for a Premier League win, and although the contest was much closer this time around, they rarely looked like emerging with anything to show for their efforts as they slipped to just a second defeat in eight outings.
Bruce had admitted after last weekend's drab 0-0 draw at Fulham that the visiting supporters might have been better off spending the day Christmas shopping, and he was in no mood to witness a repeat on home soil.
But with Wanderers heading for Wearside sitting in sixth place, one ahead of their hosts, he knew they would have to be at their best to get something from the game.
The sides largely cancelled each other out during the opening half-hour with the visitors enjoying slightly the better of what chances there were.
Martin Petrov warmed the returning Gordon's hands with a seventh-minute snapshot, but the £9million man had to be at his best to deny Wanderers two minutes later.
Matthew Taylor managed to forced his way past Nedum Onuoha on the left to pull the ball back from the goal-line, and Kevin Davies seemed certain to score until the Scotland international blocked his low drive with a foot.
Bent was guilty of a bad miss at the other end four minutes later with he and strike-partner Asamoah Gyan seeing little of the ball.
However, as the half-hour mark came and went, Sunderland stepped up a gear despite the loss of central defender John Mensah to an ankle injury, and to good effect.
It took a fine save from Jaaskelainen to keep out Gyan's 31st-minute shot after it clipped Gary Cahill on its way towards goal after Lee Cattermole had played the striker in over the top.
But the respite was short-lived and Sunderland took the lead seconds later.
Bent did well to control a deep cross beyond the far post and although the keeper parried his shot, Welbeck arrived right on cue to head the rebound home.
Cahill found his way into referee Chris Foy's notebook for a crude 34th-minute challenge from behind on Bent, and he was soon followed by Cattermole, but the Bolton defender was very nearly instrumental in dragging his side back into the game on the stroke of half-time.
He got up well at the far post to head Taylor's corner back across goal, and Knight seemed certain to score until Gordon somehow managed to turn the ball over the crossbar.
Cattermole enjoyed some good fortune four minutes after the restart when, having been booked before the break for a foul on Stuart Holden, he escaped a second and his third red card of the season for a late challenge on Johan Elmander.
The Bolton striker tested Gordon with a 52nd-minute snapshot, but Sunderland might have increased their lead seconds later when Welbeck met Jordan Henderson's free-kick with a downward header which Jaaskelainen claimed just before Gyan could get to it.
They came agonisingly close to a second goal with 20 minutes remaining as the visitors started to wilt.
Bent miscontrolled Henderson's cross with just Jaaskelainen to beat, but the ball ran to Gyan, who looked up before pulling it back for the fast arriving Welbeck.
The 20-year-old opened up his body to slide a side-footed shot across the keeper towards the far post, but saw the ball clip the outside of the upright and drop wide.
Bolton might have snatched a point four minutes from time when the ball dropped to Cahill 15 yards out, but his goalbound effort was well blocked by Anton Ferdinand with Sunderland refusing to release their grip on the points.
Substitute Bolo Zenden might have increased their lead at the death, but headed wide from Henderson's cross, but Ivan Klasnic was guilty of a worse miss in stoppage time as the visitors' hopes faded.