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.
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.
Source: PA
Source: PA