Austria international Arnautovic has agreed a five-year deal with the Hammers, who will pay a guaranteed £20million with a further £5million payable on a conditional basis.
West Ham fans feeling confident about Arnautovic joining the club can back the 28-year-old at 200/1 with Sky Bet to finish the season the season as the Premier League's top goalscorer.
The forward, who joins Joe Hart and Pablo Zabaleta in signing for Slaven Bilic's side this summer, told whufc.com: "It feels special to me to be a West Ham player. Everyone knows that West Ham is a big club, with big history and I'm happy to be a part of it now. I can't wait to get started.
"I can only say it is a massive club with a lot of fans. They are crazy for football and this is what I like. The club is still growing, getting better and better and that's why I'm here."
Hammers joint-chairman David Sullivan, who agreed a deal that should trump the £20.5million paid to Swansea for Andre Ayew last year, added: "We're delighted to have completed Marko's transfer from Stoke City.
"Alongside Pablo Zabaleta and Joe Hart we have brought in three players with vast Premier League experience this summer, and that was one of our key targets as we look ahead to the new season.
"Marko will bring plenty of quality to our front line and I'm excited to see him in a claret and blue shirt."
Arnautovic, who has won 62 caps for his country, joined the Potters from Werder Bremen in 2013, scoring 26 goals in 145 appearances for the club.
Stoke said in a brief statement: "The club would like to take this opportunity to thank Marko for his efforts and wish him well for the future."
Potters boss Mark Hughes has already let veterans Jon Walters and Glenn Whelan leave in pre-season, adding Darren Fletcher, Josh Tymon and Chelsea loanee Kurt Zouma.
