The Hammers have been looking to strengthen their forward line, with the constant rumours of a possible exit for exiled talisman Dimitri Payet.
They made a move for the Scottish international at the beginning of the month, but their initial £3 million bid was rejected by the Tigers.
Snodgrass was left out of the side for Hull City’s 2-1 win over Manchester United on Thursday night, and completed his medical at Rush Green on Friday.
The Scotsman follows on from Jose Fonte and Nathan Holland, in a deal is believed to be worth £10.2 million and will keep him at the club until the summer of 2020.
Delighted to join a massive club
Snodgrass alongside Fonte, is expected to make his debut on Wednesday against Manchester City, and the midfielder expressed his delight at signing for the club.
“I'm delighted because this is a massive Club with great tradition,” Snodgrass told whufc.com. "This is the sort of Club which speaks for itself."
"The fans who come here and support week-in week-out," he stated. "And moving to and selling out this new stadium is terrific and I want to be part of it."
"The owners and the manager are trying to build something here," the Scotsman stressed. "I just can't wait to get started."
" I feel this is a Club with real ambition with the new stadium," the 29-year-old admitted. "Great players and a manager of his calibre at Premier League level."
"The Club has had a few good weeks with results and if I can add to that great," Snodgrass concluded. "I'm just looking forward to meeting my new teammates and getting started."
West Ham United will take on Manchester City at the London Stadium on Wednesday, February 1 with kick-off at 3pm GMT.