Still reeling from their 7-0 hammering at Manchester City last weekend, Chris Hughton's struggling side looked to be heading for another defeat when Ravel Morrison gave West Ham a first-half lead with his third Premier League goal.
However, Jussi Jaaskelainen dropped the ball and then upended Gary Hooper to allow the former Celtic striker to score his first Premier League goal from the penalty spot.
Then Snodgrass, back in the side after recovering from concussion, sealed Norwich's first win in five top-flight games with his first goal of the season after 72 minutes to ease the pressure on Hughton.
Leroy Fer made sure of the points with a clinical left-foot finish in stoppage time as Norwich secured their first victory over the London club since 1994, with West Ham paying the price for a lack of a cutting edge once again and now without a win in four.
Snodgrass, Johan Elmander and defender Ryan Bennett all came into the side at the expense of Bradley Johnson, Sebastien Bassong and Steven Whittaker following the drubbing at City.
James Tomkins replaced the injured Winston Reid in the West Ham defence, while Joe Cole was preferred to Matt Jarvis in midfield.
Norwich could have been in front after just two minutes when Snodgrass picked out Anthony Pilkington, but Jaaskelainen did well to keep out the winger's left-foot strike.
A lack of goals has been a problem for Sam Allardyce's West Ham this season, but they showed a real threat going forward and Norwich keeper John Ruddy had to be alert to keep a Kevin Nolan shot out with his legs after 11 minutes.
Norwich were on the back foot and Guy Demel's powerful header rattled the crossbar a minute later, then Ruddy denied Nolan once again.
The visitors, though, were in front 32 minutes in when Nolan failed to get anything on Razvan Rat's cross, but the West Ham skipper kept the ball in and picked out the onrushing Morrison, who was left with a simple finish.
Norwich had looked short of ideas going forward, but were gifted an equaliser eight minutes into the second half when Jaaskelainen inexplicably spilled the ball and was adjudged to have upended Hooper as he attempted to retrieve it.
Referee Jon Moss pointed to the spot and Hooper drilled his penalty down the middle to open his account in the Premier League.
Jaaskelainen could only watch on as Jonny Howson's long-range strike thumped against the bar 20 minutes from time as both sides strived for a winner.
Snodgrass then produced a moment of class to put Norwich in front with a free-kick from around 20 yards out that looped over the wall and into the bottom-left corner of the net.
Midfielder Fer then wrapped up the win in stoppage time when he raced through and slotted home with his left foot to ensure Norwich leapfrog West Ham in the table.
