Everton went into the game on the back of last weekend's shock FA Cup defeat at home by Wigan Athletic, but produced the perfect response against the Premier League champions.
Goals from Leon Osman and Nikica Jelavic secured the points as Everton played the final half-hour with 10 men after Steven Pienaar was sent off for a second bookable offence.
Moyes felt Pienaar was unfortunate to be dismissed, while Kevin Mirallas had a goal ruled out for offside in the first half and Marouane Fellaini was blocked by Matija Nastasic in the penalty area.
"We were very good and have been for the majority of the season," Moyes told Sky Sports. "There have been one or two dips here and there, but that was certainly more like us today.
"It was top notch. We were playing the champions of England at this present time, who are second in the league and you're not always going to outpass them or outplay them.
"When they did get the chance to do that, when we had 10 men, we had to make sure we were hard to beat and hard to play through which we did.
"I thought we played really well in the first half. I thought we had a goal wrongly disallowed, I think there was a penalty kick and I don't think it's a sending off."