Matt Jarvis had given West Ham the lead in the 40th minute at the Emirates Stadium with a close-range header, but they could not keep their lead intact until the break.
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Stewart Downing's weak clearance gifted the ball to Santi Cazorla, who presented Lukas Podolski the chance to equalise with a powerful low shot.
Another poor clearance led to Arsenal's second goal through Olivier Giroud which ended the game in Allardyce's eyes as Podolski sealed the result late on.
"We shot ourselves in the foot with the first two goals and I think that when you do that it's always going to be difficult when you fall behind at the Emirates," the West Ham boss told reporters.
"When you do so much to get the first goal, not getting in [to half time] at 1-0 was a big disappointment.
"The first 12-15 minutes of the second half we were looking comfortable, but the second goal sickened me because the performance meant nothing after that.
"In the end you try and fight back, but we didn't get enough opportunities, we didn't create enough to get to 2-2.
"It's disappointing that down to our basic errors Arsenal have taken advantage and scored two goals, and all our hard work has been wasted tonight."
Defeat leaves West Ham looking over their shoulders with the bottom four only seven points behind them.