Demba Ba's superb finish on 49 minutes proved decisive, as did Petr Cech's incredible point-blank save from Chicharito's close-range header soon after.
United had chances to get back into the match, but Sir Alex felt Ba's goal changed the flow of the game in Chelsea's favour.
"We were comfortable in the first half and I couldn't see them getting into the game or being any great threat to us. But the goal, as it always does, changed the game," the boss told MUTV and ITV1.
"We spoke about Ba and the experience we had with him a couple of years ago at Newcastle when he scored exactly the same type of goal. We were caught napping to be honest. It wasn't a good start for us right at the beginning of the second half.
"After that I think we lacked the composure to win the game and create proper openings. We had a lot of possession but didn't make the best use of it and they caused us quite a few problems on the counter-attack.
"They've got several players like Hazard, Oscar and Mata who sleep in little pockets and can cause you bother on the counter. Fortunately we didn't lose another goal, but I couldn't really see us scoring either.
"Robin van Persie had a fantastic chance and should have scored, but he put it over the bar. We had a couple of headers past the post and Cech made a fantastic save from Chicharito. But we just lacked composure when we had to chase the game and didn't play as well as we should have in the second