Allardyce said that he was pleased with how his team reacted after going a goal down and feels that they should have gone on to win.
"We could have finished them off with more clinical finishing, but there is another point on the board for us and we have to settle for that," BBC Sport quotes Allardyce as saying.
"I liked the way we sustained the pressure on Stoke, the belief and the spirit was not dented by the early goal. I think everyone in the second half raised their game, took Stoke to the wire and I think everyone would say we deserved the three points."
The Hammers remain in seventh place, but are now level on points with Arsenal.