Joey O'Brien scored the equaliser for the Hammers but the 58-year-old saw more chances spurned, with on-loan striker Andy Carroll still yet to get off the mark for his temporary club.
"In the end we were disappointed we didn't win it," Allardyce told the press after the match. "In the second half, when you put that much pressure on, you've got to try and be a bit more ruthless in front of goal and we probably could have won it.
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"Finishing the ball in the box or creating chances is hard but scoring chances is the most difficult thing in this league and the quality of play and composure is hugely expensive. If you want to buy the top ones you pay £50 million for Fernando Torres, or £24m for Robin van Persie, who was in the last year of his contract.
"At the moment we've only got Kevin Nolan who's doing it on a regular basis [with five goals] - and Kevin will be disappointed that he didn't score that chance at the start of the second half."
Asked about Carroll, Allardyce replied: "Andy's not to get too frustrated by the fact that he's not scoring at the moment. In fact, today, he didn't get too much of a chance to score either.
"I think, sooner or later, if we continue to work on him making runs in the box and on the service we give, then eventually he will score.
"He wants to score a goal and he wants me to leave him on for 90 minutes but, at the end of the day, he put that much effort in, like all the players, there's some fresh players on the bench. He'll be disappointed, I know."
|"Sooner or later, if we continue to work on him making runs in the box and on the service we give, then eventually he will score"
- Sam Allardyce on Andy Carroll
Stoke boss Tony Pulis appeared content with the result, stating: "I thought we played well first half. We started very sloppy second half - in the first 10 minutes, the amount of times we gave the ball away, and then we just invited West Ham onto us and they scored a goal.
"From that point onwards, we had to show a lot of resilience and a lot of determination as a group not to concede a second."