After suffering the humiliation of a 7-0 defeat at Manchester City last weekend, the Canaries seemed bereft of confidence in the opening half against a West Ham side who led at the interval courtesy of a Ravel Morrison strike.
But Norwich rallied in the second half with a Snodgrass free-kick giving Chris Hughton's side the lead after Gary Hooper had levelled through a penalty.
Leroy Fer's injury-time strike added the gloss to a vital win for Norwich, with Snodgrass revealing the players were aware they were not playing well enough during the first 45 minutes.
"In the first half the fans were on our backs a little bit," he 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."
Scotland international Snodgrass missed the thrashing at the Etihad Stadium after suffering a concussion in a 4-0 Capital One Cup defeat at Manchester United four days before.
But he shared in the agony of the Canaries' largest ever Premier League defeat and hailed the performance of forward Johan Elmander as a prime example of how the players wanted to react.
"It is just the highs and lows of the Premier League," said Snodgrass.
"I think the way we went about our business in the second half shows you the character and the belief among these players.
We felt as low as we could be after a 7-0 defeat. But we need characters, we need men at this football club and we showed that in the second half.
"Our unsung hero was Johan Elmander, I thought he chased every ball. He didn't get the goal he deserved but sometimes you get that as a striker. But he put his body on the line for the team and that is what it takes."
Norwich started the day in the bottom three but the win took them above the Hammers and up to 15th place in the table.
The visitors had dominated the first half and, as well as Morrison's opener, had forced John Ruddy into two smart saves with Guy Demel also heading against the crossbar.
With such a commanding display, full-back Razvan Rat was at a loss to explain how Sam Allardyce's side ultimately slipped to defeat.
"It's amazing how we lost it," the Romania captain told West Ham's official website.
"It is difficult to think how we lost the game after the first half in which we played very well, had a lot of chances and controlled the game.
"It should have been all over by half-time. If we had scored the second goal and maybe a third one, it would have been hard for Norwich. But I think this is our problem - we created a lot of chances but we hit the target not too often. We need to improve on that."