The Hammers took the lead through Mohamed Diame after 21 minutes but were pegged back by strikes from Olivier Giroud, Theo Walcott and Santi Cazorla.
And the Englishman highlighted chances missed by Kevin Nolan and Andy Carroll with the scores still level as crucial points in the match.
|11/2||West Ham are 11/2 to be relegated with William Hill|
"We did with the first one - Mo's goal was brilliant - but then there was one just before half-time when Kevin couldn't quite dig it out from Mark Noble's free-kick, and then in the second half Kevin has had two golden chances.
"Then Andy has had his header...We'll look back at those and think that, at one-one, that would have put us in front.
"That's Arsenal for you. If you don't take advantage of your chances when you get them then they will punish you and they certainly punished us towards the end of the game."
The 57-year-old also rued the fact he could be without the services of Ricardo Vaz Te, Guy Demel and Winston Reid for a prolonged period, after all three picked up injuries during the game.
"I don't think we'll have them back in two weeks [after the international break]," added Allardyce.
"I think they'll be out for a little bit longer than that by the looks of their injuries. Certainly Ricardo is a big blow because it looks like a dislocated shoulder and that's anything from eight to twelve weeks."