The hosts were outplayed by their visitors throughout the first period as a neat corner routine allowed Jon Walters to fire Stoke ahead, but came back strongly in the second period to earn a point through Joey O'Brien's effort.
Allardyce told Sky Sports afterwards: "I thought we deserved three (points) in the end, over the 90 minutes. I thought Stoke were better than us in the first half.
"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.
"You can see ours (corner routine) too when Gary (O'Neil) was about five or six inches wide of the far post. It nearly resulted in a goal - theirs did and I guess that ended up being the difference today between winning and draw.
"There were other chances. 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."