Stoke, however, hinted at their first victory in 10 games and only their second on the road all season when Xherdan Shaqiri's 79th-minute effort was spilled by Three Lions hopeful Joe Hart and Crouch thundered home the rebound from close range.
Manager Paul Lambert has no further injury concerns to contend with for the trip to the London Stadium.
Andy Carroll scored a 90th-minute equalizer for West Ham to draw with relegation-threatened Stoke 1-1 in the English Premier League and come to teammate Joe Hart's rescue after the goalkeeper's latest error on Monday.
Shortly after Moritz Bauer produced a brilliant run and cross for Stoke striker Mame Biram Diouf, who could only scuff the ball over the cross bar.
With the tempo rising, Arnautovic hit the target with a powerful near-post drive only to find Butland in the way, the England goalkeeper in flawless position to block the effort with his face. On the first occasion Marko Arnautovic was offside when he headed home - and then he was interfering with play from Edimilson Fernandes' shot.
Diouf then missed Stoke's clearest chance, turning a Bauer pullback over the crossbar with a first-time strike that wasted a unsafe break.
The other subplot to the evening was that this was Michael Oliver's first game as a referee since the now-infamous Champions League tie between Juventus and Real Madrid, Gianluigi Buffon's red card and all the unpleasantness that has followed.
Alex Oxlade-Chamberlain has been superb for Liverpool this season following his move from Arsenal in the summer of 2017, and even West Ham United vice-chairman Karren Brady has been impressed, according to her comments in The Sun.
Despite dropping two crucial points, Stoke City manager Paul Lambert said he still took a number of positives from the result.
The former England global quickly came up with a massive moment for his side, showing attacking instinct to ram home the loose ball after Hart somehow let Shaqiri's strike slip out of his grasp.
Javier Hernandez soon rifled home what might have been a Hammers victor, but Carroll had handballed and no goal was given.
Stoke City have not won a game since 20 January this year - a 2-0 win over Huddersfield Town.
"I'm keeping my target high and I'm hoping to win ten games". I couldn't ask for any more from my players.
But Carroll's goal keeps the Potters in 19th place, with just four games to go.