Everton dominated and Jonjoe Kenny struck his effort against the woodwork.
The closest that West Ham, in their first Division 1 match since promotion, came to a goal was Antonio's shot, which went narrowly wide.
Conor Coventry was shown a second yellow in stoppage time but West Ham held on for a point.
Their next match is at Swindon Town in the Checkatrade Trophy, with Everton travelling to Notts County.
"He was first class in terms of his attitude and the way he approached the game."https://t.co/v0gPuEt7rr— West Ham United (@WestHamUtd) August 11, 2017