The Norfolk outfit had struggled for form this season, ramping up the pressure on manager Chris Hughton, but recorded a hard-fought victory at the Britannia Stadium to move clear of the relegation zone.
"With performances like that, I don't know how anyone can criticise us," Snodgrass told the Norwich Evening News.
|3/1||Norwich are 3/1 with Coral to be relegated
The Scotland international appreciates that expectations have been raised following a summer of big spending at Carrow Road, but is convinced the side can deal with the added pressure.
"The fans are right behind the manager - they're a different class here," he continued.
"They felt frustrated that we had so many chances against Aston Villa the previous week and didn't take them, but there's no better way to put things right than to roll your sleeves up and pull off a result like that.
"He has spent maybe £27million on players and the manager will tell you himself that there is an added pressure when you have a bigger kitty to spend.
"We're in the Premier League and there's pressure. I'll be honest and say that if you can't handle that pressure, there's no point in being here.
"In your job, every day, you put pressure on yourself. Even on the training ground, we're putting pressure on ourselves to get better and to work on our weaknesses.
"The manager puts that pressure on us - and the fans put the pressure on the players and the manager. It's not just the manager. We're all in it together."