Winston Reid and James Tomkins both let a through-ball go for the first goal as Sergio Aguero ran on to the pass and found the bottom corner, while the Argentine striker was also left completely unmarked from a free kick for the second goal.
Allardyce admits that the result was not what he wanted on his 59th birthday, but did praise the efforts of goalkeeper Jussi Jaaskelainen in an otherwise disappointing first half.
"It's a miserable birthday for me. The only thing you want for your birthday is a good, positive result, but we didn't get that and unfortunately, when you play a side of Manchester City's quality, we need to hit top form," Allardyce told the club's official website.
"Man City were controlling the tempo and then we made, unfortunately, a howler of an error for the first goal. It was just a straight little ball down the middle and unfortunately neither of our centre-halves dealt with it, which is very unusual for them. And you don't give Aguero a one-on-one with the goalie. Very rarely does he miss and he puts it in.
"Not only did we do that, we did exactly the wrong thing on a free-kick six minutes into the second half. So the first half was very disappointing, apart from Jussi, who kept us in the game with some outstanding saves."
The Hammers now sit 14th in the Premier League table and will face Swansea City in their next match.