Aside from a fine free-kick from Leighton Baines which thumped the visitors' crossbar, Everton rarely looked like piercing a resolute Reds defence, while Chicharito capped the game's move of outstanding quality to bag all three points.
"It was a hard game," Sir Alex told MUTV. "We expect a hard game every time we come to Everton, in fairness. The crowd get behind them and they show great enthusiasm.
"In the first half in particular we played very, very well. Second half we defended well and that was the pleasing aspect for me. Winning 1-0 - I'll take half a dozen of those anytime because that's championship form when you get those kind of results.
"We had probably the best chances in the game. All their attempts were crosses and we managed to get most of them away. Really David [De Gea] has not had anything to do in terms of saves.
"They were unlucky when they hit the bar, it was a fantastic free-kick, so we rode our luck on that one. I thought we just battled in the second half and got the result we wanted."
With last weekend's exposure of United's attacking instincts still fresh in the mind, the Boss was reassured by his side's core strength at Goodison Park, with Nemanja Vidic outstanding and Wayne Rooney deployed centrally.
"Everyone's telling us the stats about chances made against us," he said. "We have to reduce that, If we're going to win the league we have to reduce those types of statistics against us. Today we were very solid.