Welbeck netted in the first half and goalkeeper Craig Gordon then made a point-blank save to deny Zat Knight on the stroke of half-time.
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Welbeck, on loan from Manchester United at the Stadium of Light, told Sky Sports 1: 'That was my aim when I came here to get games. I'm just looking forward to carrying on my form.
'Every game in the Premier League is going to be tough but to get the three points and go ahead of Bolton (in the table) is great.
'Everyone's working well together and the team spirit is very high.'
Gordon admitted his stop from Knight had been a moment of rare action for him in the match.
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'It's probably the only real save I have had in the game but I managed to get a real solid hand to it,' he said.
'It was a good save and it was important at the end that we got three points and that's what makes making saves like that even more important.'
Ivan Klasnic's late miss made sure Sunderland were not denied the win and Gordon admitted: 'We got a bit of a life there but we will take the good luck when it comes.
'It's important for the team to keep clean sheets and keep grinding out 1-0 wins.'
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Explore more:People: Ivan Klasnic, Zat Knight, Craig Gordon, Danny Welbeck