Sir Alex thinks it's going to be a long season for United with Manchester City only a point behind the Reds in the table.
United scored after 33 seconds through Robin van Persie but it became a nervous night against West Ham on Wednesday with the victory not assured until the final whistle.
Manchester City's 2-0 win at Wigan Athletic keeps them in contention with the derby taking place at the Etihad Stadium on 9 December after United's trip to Reading at the weekend.
"We've got the three points and are still ahead in the league," the boss told MUTV. "Chelsea drawing tonight means it draws us away from them a little bit further. City are still on our tails so it's going to be a long season."
Sir Alex was pleased to keep a clean sheet and praised defenders Jonny Evans, Chris Smalling and Rafael for their stellar performances.
"It was a hard night and we played a lot of good football but we never quite finished them off," said the manager. "West Ham fought really hard and made it difficult for us.
"It was a great start and hopefully you go on from such a great start but we never quite finished it off.
"They had skirmishes around the box and played high balls in there but you've got to defend them really well. They had a few blocked shots and that kind of thing but, in the main, we did defend very well. Evans, Smalling and Rafael, in particular, had fantastic nights.
"Rafael is maturing. He's 22 years of age but came here as a young kid and was in the first-team at 18. He made rash mistakes because he's such an enthusiastic boy but he's maturing and is learning from that."
Sir Alex refused to accept that a lack of a width was an issue with no out-and-out wingers in the starting line-up, although Ashley Young and Danny Welbeck played on the flanks when coming on as substitutes.