|14/1||England are 14/1 with William Hill to win Euro 2016|
Welbeck broke the deadlock in the 58th minute, the forward finishing Raheem Sterling's excellent low delivery from the left at the back post.
Gary Cahill made a vital goalline block from Josip Drmic with 20 minutes remaining, and Welbeck made the most of England's reprieve deep into stoppage time, keeping his cool when one-on-one with Yann Sommer after being played in by Rickie Lambert.
Speaking after the game, Hodgson commended his team's performance and felt Welbeck's breakthrough was the key to success.
"I'm just pleased to see the boys play well the way we approached the game from the first minute was exactly what we wanted," he told ITV.
"We got our reward for once, we made hard work of it, there were plenty of chances before Danny's goal but it was a good performance against one of the best teams in the group.
"In the first half they were a bit content to let us dominate and we pressurised the ball well.
"You need that first goal to open people out, then they started to come at us. From then I thought we looked dangerous and we had several chances to aggravate the scoreline before Danny Welbeck's second."
Despite the win, Hodgson insists England can play better, adding: "I still think there are areas we have to improve, you can't expect players playing in their early games or even their debut not to make mistakes.
"There are areas we can work on and we have two years to do that."