The United manager was disappointed with his side's lack of chances at the Etihad Stadium and also criticised Roberto Mancini's conduct towards officials on the touchline
Vincent Kompany scored the only goal of the match on Monday evening to help City leapfrog the champions and reach top spot on goal difference with just two games to go.
Sir Alex conceded that the advantage could prove significant with the two sides locked neck-and-neck at the summit of the league table after United's derby defeat.
"They are in the driving seat now," he told Sky Sports. "They only need to win two games of football.
"It's not over yet, of course – as long as there's games still to play, it's not over. We've both got the same points, but they have an eight-goal advantage and that's a great advantage to have at this time in the season."
Tensions boiled over during the second half as Sir Alex clashed with counterpart Roberto Mancini on the touchline.
The United boss revealed he had been unhappy with the Italian's conduct towards the officials and hinted that City had received more beneficial decisions than United during the match.
"He [Mancini] was badgering the officials all game," Sir Alex complained.
"The minute I go out there, he's up again. He's been complaining about referees all week – he can't be complaining tonight though, that's for sure."
The 70-year-old expressed his disappointment at the lack of chances his side had created during the match, with Joe Hart remaining largely untested in the City goal.
United started the game positively but fell on to the back foot as the hosts established control of possession before Kompany headed home the crucial goal in first-half added time.
"It was the kind of game we expected," Sir Alex reflected. "There were not a lot of goal chances - I think David de Gea made the only save of the match.
"We are a bit disappointed we never tested their goalkeeper – our crossing was poor. They were more of a threat, simply because they had more action around the box. We had control over some parts of the game but never enough to do any damage.
"If you concede a goal from a set-piece at this level of football you've only got yourself to blame.
"Nothing was happening in the first half really. We started well; for 10 or 15 minutes we looked like we were on top.
"As the game wore on they had lots of possession outside the box without really doing anything. Then they get a corner kick and score so that was a really bad time to concede just before half-time."
United blew a 4-2 lead against Everton on Saturday and Sir Alex recognised that the result against the Toffees could come back to haunt them.
"You can look back at any season and say that was a bad result, but sometimes that doesn't affect you," he added. "But I think Saturday against Everton was the worst game we have had in this run."