The Hammers drew 1-1 at Upton Park, with Joey O’Brien equalising following Jon Walters’ first-half opener.
However, Allardyce was adamant after the game that his team did enough to overcome their opponents.
“I thought we deserved three (points) in the end, over the 90 minutes. I thought Stoke were better than us in the first half,” he told SkySports.
“We had to recover from an extremely good goal from Jon Walters. It was well-worked and an extremely good finish. Having to come from 1-0 down in any Premier League game is difficult.
“The way we said to the full-backs at half-time: ‘Let’s get forward more’ was key. Kevin Nolan had a great chance from George McCartney and that set the stall out for us. Then we scored from another chance.
“There were several other chances that we had. We needed to finish them off because we created enough chances to win it, but we move on with another point on the board reasonably happy.
However, the ex-Bolton and Blackburn coach was happy with the character his side showed after going behind.
“We rarely let them out of their own half and that really pleased me after the knock-back of going 1-0 down. I’m really pleased with the spirit and the belief,” he added.