The Hammers fell behind at the Etihad Stadium in the 18th minute when James Collin attempted to clear Jesus Navas' pass only to see the ball lob his own goalkeeper Adrian and go in off the woodwork.
18 minutes later and West Ham midfielder Stewart Downing gave up the ball to Yaya Toure who played in Sergio Aguero for City's second goal which prompted the following response from a despondent Allardyce.
"The disappointing thing firstly was the goals were gifted to Manchester City and, secondly, the amount of possession we allowed ourselves to lose," said Allardyce.
"The two mistakes we made for the goals, one was pretty bizarre and the other unexpected from a player as experienced as Stewart Downing.
"We had the resilience and determination to try to fight back and in the second half, eventually changing the system got us back playing better going forward.