Pedro converted an early penalty after being brought down by Kepp.
Then, Welbeck sealed the victory with a brilliant finish minutes after coming off the bench.
Manager Fabian Hurzeler stated post-game: "In the first half we won a lot of duels and second balls and we deserved to lead. We could easily have led by two goals at half-time but you can’t change that and we had to refocus and that’s what we tried to do. But in the second half we suffered.
"The goal came out of nowhere and was an amazing shot and after that we had to stick together. We weren’t that aggressive, and Bournemouth controlled the game. The momentum changed a little bit so it was a difficult situation for us and in the end we were a lucky winner.
"But we did the basics right. To win a Premier League game is so tough. We tried to outrun them and win our personal duels and that was the key to winning the game. Everyone has a responsibility to defend and I saw that from us as a team - we wanted to defend and attack together. We pushed the limits tonight. It wasn’t the most beautiful game from us but we were very effective.”