Having fallen behind to Romelu Lukaku's opener the visitors fought their way back into the match at Goodison Park, and got their reward when substitute Mauro Zarate levelled with the help of a deflection.
West Ham continued to press and Morgan Amalfitano, Matt Jarvis and Andy Carroll all wasted good chances before Leon Osman gave Everton a 2-1 win.
Allardyce rued the lack of a clinical touch inside the penalty area, and felt the least his side deserved was a draw.
"I am hugely disappointed," he said. "When we've been in the position to get the second goal, and it was looking possible, one punt up the middle was a hugely disappointing way to concede a second goal.
"Defensively that was the only disappointment I can say today, unfortunately it cost us a goal.
"[But] our finishing was not good enough, and because of that we've ended up losing."
Allardyce also questioned the legitimacy of Everton's opener, the 60-year-old adamant Lukaku was in an offside position when he fired past Adrian.
"That decision baffled me," he added. "It is blocked by [Winston] Reid and falls to Lukaku who is offside.
"I have had one [an explanation] from the referee, and I will discuss it with PGMOL [Professional Game Match Officials] next week."
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