Robin van Persie netted the fastest league goal so far this season with a deflected shot in the opening minute but the Red Devils were unable to convert any more of their chances.
Sir Alex told Sky Sports News after the game: "West Ham are renowned for making it difficult for you - they're a very dogged team to break down - but we got a great start and I thought from there we could have pushed on.
"Some of our football was very good at times but we never finished them off."
Asked if it felt unusual for his side to score the first goal in a game, after coming from behind to win six times already this term, the Scot quipped: "Yes - and to keep a clean sheet. That's Christmas time!
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"So we do have plus points tonight, particularly defending. You have to against them because you have to against them because they're such a handful.
"We had opportunities to finish them off with some good football but you've got to give credit to West Ham, they did really fight hard. They're a very well organised, big powerful team. I'm pleased with the result."
The win sees United retain their place at the top of the Premier League, one point above Manchester City and Sir Alex added: "I think that if we can keep clean sheets more often it will give us a good foundation with which to go for the title."