Lambert said afterwards: "It is a blow because we were so close to winning".
The hosts are expected to be without the services of Pedro Obiang, Winston Reid, Sam Byram, Andy Carroll and Michail Antonio.
Stoke will again be without forward Maxim Choupo-Moting, who is not yet ready to return from a groin injury.
The draw leaves Stoke on 28 points from 34 games, level with Southampton and five behind 17th placed Swansea City who both have a game in hand, while West Ham are 14th with 35 points.
"We didn't go out there with it really planned".
"Last week after he made fantastic saves at Chelsea, I said to him: "Well done".
"I think if the team had been playing this way from August, I wouldn't be here and they wouldn't be in the position they are in".
"I came in at a late time, but I absolutely love it and we are well in the fight".
"We've still got four games so no-one is safe".
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Butland didn't know much about Arnautovic's angled shot that struck the Stoke keeper in the face on 36 minutes after Pablo Zabaleta had kept the ball in play on the right, and Arthur Masuaku flashed a long-range effort wide of the right post moments later.
"All managers and players and teams need a little bit of Lady Luck to have success and it has evaded us, not just in this one but in a few other games".
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The Hammers are aiming to complete their first league double over Stoke since the Potters' first season in the Premier League in 2008-09.
It's been another frustrating season for West Ham, but things have been looking up recently as they bid to avoid relegation from the Premier League. They will aim to make it two in a row at home when second-from-bottom Stoke City visit London Stadium on Monday. With AP Photos.Andy Carroll scores a 90th-minute equalizer soon after coming off the bench to earn West Ham a 1-1 draw against Stoke, which is denied a vital win in its bid to escape relegation from the English Premier League.
But another substitute would yet have a decisive impact on the result, Carroll superbly turning in a cross from Aaron Cresswell to ensure the spoils were shared at London Stadium.
They are one loss short of reaching 1,000 defeats in the top flight and would become the 16th club to reach this landmark.
Teenager stabbed to death in Forest Gate
An London Ambulance Service spokesman added: âWe were called to Head Street to reports of a stabbing. In relation to the Brixton incident, police arrested a man in his 20s on suspicion of murder.