The Potters took a 13th-minute lead when Republic of Ireland internationals Glenn Whelan and Jonathan Walters combined from a set-piece, with the latter finishing past Jussi Jääskeläinen.
However, West Ham were much-improved after the interval, and Joey O’Brien managed to get on the end of Gary O’Neill’s low cross to salvage a point for Allardyce’s side, with Kevin Nolan missing late chances to snatch a win.
“Well I thought we deserved three points in the end,” said the West Ham boss.
“Stoke were slightly better than us in the first half and we had to recover from what was an extremely good goal from Walters.
“Having to come from 1-0 behind in any Premier League game is difficult, but the way we started the second half by pushing the two full-backs on and getting forward set the stall for us.
“It was a ball down the right and Joey O’Brien scored, but we had several other chances and if we had the clinical finish then we could have won it because we created enough opportunities.
“But having come from 1-0 down, it’s another point on the board and we move on reasonably happy.”
Meanwhile, Pulis praised Stoke’s resilience to temper a strong second-half display from the Hammers in front of vociferous home support at Upton Park.
“It was a game of two halves,” Pulis said. “The worst thing that could happen to us was half-time.
“We played really well first half and we were a bit disappointed not to come in at the interval more than 1-0 up.
“We showed a lot of resilience in the second half and we had to defend for our lives.”