The Canaries relieved some of the pressure that had been mounting on their boss, coming from behind to secure a hard-fought home win over the east Londoners
Fer's last-gasp third added to Gary Hooper's 54th-minute penalty and Snodgrass's wonderfully-struck free kick to eventually render Ravel Morrison's predatory close-range opener worthless, with the hosts recovering from their insipid first-half showing - and a one-goal deficit - to hang on to the victory.
Still reeling from a 7-0 hammering at Manchester City last weekend, Hughton's struggling side looked to be heading for another defeat when Morrison gave West Ham a first-half lead with his third Premier League goal.
However, Jussi Jaaskelainen dropped the ball and then upended 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-footed finish in stoppage time as Norwich secured its first victory over the London club since 1994, with West Ham now without a win in four.
Hughton made three changes to last weekend's side, with 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 defense, while Joe Cole was preferred to Matt Jarvis in midfield.
A lack of goals has been a problem for Sam Allardyce's West Ham this season, but it initially 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 was 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 was gifted an equalizer 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 looked 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 the win that sees Norwich leapfrog West Ham in the table.