Ravel Morrison gave the Hammers a first-half lead with his fifth goal of the season, which was cancelled out by Gary Hooper's equaliser 10 minutes after the break.
But Snodgrass fired home a free-kick midway through the second period before Leroy Fer struck in injury-time as Norwich claimed the three points at Carrow Road.
Snodgrass was pleased with how he and his team-mates bounced back from last weekend's 7-0 defeat against Manchester City.
He told Sky Sports 1: "After the 7-0 defeat to Man City last weekend it just showed the character of these players and everything about us...fantastic club, fantastic fans and a fantastic spirit, and we needed that result.
"In the first half we didn't get going and we changed our shape and went to two up front.
"West Ham knocked the ball about really well and it was hard to get near them at times.
"We had a few feisty words at half-time and we needed to relax and take it as every other game, and I think we rolled our sleeves up and we deserved the victory."