The Canaries invested heavily in new talent over the summer but their Barclays Premier League campaign started slowly and they were stuck in the bottom three until Saturday's 3-1 success at home to West Ham.
Snodgrass scored the second as Gary Hooper and Leroy Fer also found the back of the net to see Norwich cancel out Ravel Morrison's goal for the Hammers.
Up until Hooper's 53rd-minute equalising penalty the Norwich fans had again started to show their displeasure and the recent speculation surrounding the future of Hughton seemed ready to start gathering yet more pace.
But Snodgrass, who was returning to the side after missing the 7-0 defeat at Manchester City through injury, feels the players have not been performing well enough for their manager.
"You all know there has been pressure on the manager," he said. "But the boys have got to take a great deal of that as well because if we're not putting the ball in the back of the net it's up to us.
"He [Hughton] can't wave a magic wand and try and make everything right, it doesn't work like that.
"We had to work on the training ground to put that right. As soon as you cross the white line your biggest aim is to try and better your opponent that's in front of you."
The Norwich players seemed to be struggling for confidence in the opening exchanges against West Ham but Snodgrass insists the fire has been burning in their bellies to correct recent short-comings.
"I think I've argued with Russell Martin and Steven Whittaker more times than I've ever argued with my partner," he said.
"But after the game we're friends. You need to let out that anger because you're doing everything you possibly can to try to win the game.
"You're not falling out with the person, you're just letting them know they could do better. Same for me. I don't go under. If you're going to fold as soon as somebody gives you criticism then there's no point in playing. I think everybody needs to do that."
The Scotland international also called for any jeers or disquiet amongst supporters to come to an end as the Canaries look to build on the win.
"I think we all have to stick together," he added.
"It is a great little family club. Support the lads, that's what we've been calling for. We need the supporters as much as they need us."
Meanwhile Hooper revealed he had been honing his penalty taking on the training ground in case he was called upon in the game.
As it transpired the summer arrival from Celtic was felled by West Ham goalkeeper Jussi Jaaskelainen and dusted himself down to convert the resulting spot-kick to justify the extra commitment.
"I was practising them yesterday and the gaffer told me I was on them," he said.
"I stepped up to take it and put it down the middle. I thought it was a good penalty under the circumstances and then we kicked on.
"I did feel the pressure a little bit - it's huge for the club, my first goal and we needed to win."