The Hammers went behind early on thanks to a debut strike from Loic Remy but the home side responded strongly to that early setback.
After spurning numerous chances, Joe Cole brought West Ham level late in the second half to ensure a share of the spoils in the London derby.
Allardyce admitted that his side's failure to convert key chances has been a source of frustration all season.
He told BBC Sport: "I don't think we have quite dominated like that this season, our outstanding performance today should have reaped more than we got.
"It's a disappointing end to a very encouraging and entertaining performance and a complete domination of the opposition. We let them break out just the once and Remy scored. We upped the tempo second half and it looked like it might be a victory but they kept getting bodies in the way, the keeper kept saving it, you just have to keep trying and hope the results go your way.
"Our Achilles heel is that we created so many chances but haven't come away with all three points."
A hard-fought point leaves West Ham in 12th place in the Premier League, seven points clear of the relegation zone.