The Canaries went into the match with one point from four league games, and having lost 7-0 to Manchester City last weekend.
Norwich looked like a team low on confidence, and after a tentative first half the home side were booed off the pitch trailing by the only goal of the game.
But Chris Hughton's men responded after the break with goals from Gary Hooper, Snodgrass and Leyroy Fer to win 3-1.
The victory moves Norwich out of the bottom three in the Premier League, and Snodgrass hopes it will prove to be a decisive result.
"In the first half the fans were on our backs a little bit," the Scotland international said.
"But that is deserved because we weren't at the races, and you take that as a player and how the fans will be on your back because they are passionate and they care, just as we do.
"The boys here, we keep going and going and we are a close bunch. It was this time last year against Arsenal that was a turning point in the season and hopefully it can be the same this time."