The Hammers enjoyed the better of the play for large parts of the opening 45 minutes as Matt Jarvis headed them in front five minutes before the break, only for Lukas Podolski to equalise moments later.
Second-half goals from Olivier Giroud and another from Podolski put the game beyond West Ham's reach and they rarely threatened after falling behind.
Allardyce was encouraged by parts of the display at Emirates Stadium, but says that they should have made more of their early chances with Mohamed Diame guilty of being wasteful in front of goal.
"I think that we'd done so much up to then to get the goal - I thought we deserved it," he told Sky Sports.
"I think that the one thing that disappointed me was that we didn't create more from our opportunities in the first half but we finally got a very good goal.
"The key was to hold onto that until half-time and then say to the lads 'going out there now, there's more pressure going to come on Arsenal and the home fans expect to beat us today'.
"It preys on their mind a little bit, they've got a bit of fatigue in the team and a few injuries.
"That would've been more pressure on them but we released the pressure on them by making two errors for both the goals and allowed them to get in front.
"Then, it's a question of whether we could create another chance in between that and then finally they got the third and that was the end of today.
"Overall, a good performance but it makes no difference at the end whether you've played well if you go out there and lose."
West Ham had a penalty appeal waved away as Bacary Sagna appeared to clip the legs of Jarvis inside the area, and Allardyce believes that it might have been given had the West Ham winger gone to ground.
"This debate is that if people stay on their feet and it's a foul, then you don't get it. And that's a fact because we've proven that over many, many years now," he said.
"While that makes life even more difficult for referees, they're paid to give fouls whether people stay on their feet or they don't and in opposite of that we continue to get slaughtered on simulation or diving and that's a great shame.
"At the end of the day, we've lost 3-1 and it's more down to our deficiency rather than the referee's deficiency."
West Ham remain seven points above the drop zone in 11th place and although slipping out of the Premier League appears highly unlikely, Allardyce is keen for a positive end to the campaign with four games remaining.
He said: "I think I've got a lot to do and we've got a lot to do between now and the end of the season in terms of making sure this season doesn't peter out and making sure that we try and get as many points as we can and try and get as many victories as we can on the board.
"Crystal Palace is going to be a massively tough game on their run at the moment. They're in top form at the minute so it's going to be one at the Boleyn on Saturday that we want to be on top form again but make sure we win if we can.
"If we can't win, make sure we don't lose. That's two on the trot we've lost and we don't want to lose any more."