Welbeck struck midway through the first-half for the 1-0 win, but was forced off late on with an apparent back injury.
Brighton boss Fabian Hurzeler said later: "I think we suffered first half. There was a lot of intensity from Newcastle. But we showed a brilliant reaction.
"In the end, I wouldn't say it was a deserved win, but my team suffered and defended together.
"Sometimes that's football. We didn't deserve to score at the time when we did. But it's great individual quality from Danny Welbeck. I am very happy with him."
On Welbeck's injury, he added: "No updates. I can't say any updates. I think he got a knee in his lower back. Let's wait until later, until tomorrow, then we'll know more.
"We defended together. In the end, defence wins Championships. We got smashed against Chelsea but we learnt from it. Not everything was perfect today, we have to be honest with ourselves. But the luck will come to you if you work for it.
"We have a lot of injuries, we always have to adjust things. But the guys are doing it brilliantly. Of course we want to focus on the process but you also need the results. For me, the result was good, the performance was OK."