However, a controversial injury-time goal evened the ledger again when referee Bobby Maddly over-ruled his linesman in awarding a goal to Ryan Wilson with the Hammers' defence vainly appealing for hand-ball.
West Ham's kitman was left with pie on his face after Arnautovic's name was misspelt on the back of his shirt - reading "Arnoutovic".
Former Premier League referee Dermot Gallagher has told Sky Sports that Callum Wilson's last-minute equaliser for Bournemouth should not have stood against West Ham, as the Cherries secured a last-gasp point. Schneiderlin moved to Goodison Park 12 months ago from Manchester United and after an initial bright start with the Blues, the Frenchman has seen his performance levels dip and his attitude questioned. So if you overrule the offside then you say that it doesn't matter that he put the ball in with his arm. I think we could have scored five or six today.
"We didn't do the things that we'd been doing the last couple of weeks and we didn't take our chances as well, the chances that we had, so it wasn't good at one end and it wasn't fantastic at the other".
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"I don't know why he wouldn't go with his linesman's decision".
Bournemouth and West Ham will both be desperate to emerge victorious against a relegation rival when the pair lock horns on Tuesday.
"It was hard to go one down after seven minutes, but we have shown character to get back in the game at 2-1".
The Hammers had come from 2-1 down to lead 3-2, thanks to Mark Arnautovic's brace after James Collins had given David Moyes' side the lead only for Nathan Ake and Dan Gosling to turn the game on its head.