Javier Hernandez soon rifled home what might have been a Hammers victor, but Carroll had handballed and no goal was given.
West Ham United are under no impression that they are safe in the Premier League but a victory against Stoke City will greatly improve their chances of staying in the top flight, defender Arthur Masuaku said.
"If we take the game as we played against Southampton or Chelsea, I think we'll win. A lot of clubs look like they will not able to get that, hopefully we do as soon as we can". "We'e playing well, we have an incredible support behind us, and I'm pretty sure come Sunday, there won't be many stadiums rocking the way ours can rock".
Moyes spoke about the impact Carroll can provide, adding: "He's been training since we were away in Miami so he has had a couple of weeks of running".
"It would have done us the world of good to get the three points but it was a great finish by Andy".
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Stoke remains 19th in the table, five points adrift of safety with four matches remaining, one of which is against UEFA Champions League semi-finalists Liverpool at Anfield, while West Ham - which saw three goals ruled out - is seven clear of the drop zone.
Stoke City manager Paul Lambert has dubbed his side's upcoming Premier League game against Burnley a must-win. "With the atmosphere in stadium, it is a hard place to come but I thought we were excellent".
David Moyes: "It was a great point at Chelsea".
But this will feel like another loss, having come so close to only a second win in his reign.
The game's first chance arrived after 30 minutes when Joao Mario played in Arnautovic, who dragged his close-range effort wide of the mark. But performance-wise it was very good.