Goals from Steven Pienaar, Marouane Fellaini and Nikica Jelavic saw the Toffees follow up Monday's home win over Manchester United in style.
Villa could only manage a late consolation from Karim El Ahmadi as they slumped to their second defeat of the campaign.
Moyes said: "The first half, I said to the players, at times it was as good as I've seen for a long time at Everton.
"We missed a few chances before we scored but we played very well.
"We have won two out of two. We've always been good finishers to a season, and we've just got to make sure it doesn't happen in reverse.
"The job is to keep us going in the right direction. If you win two, it gets you off to an initial good start but you are judged a bit more after 10 games."
Moyes will look to strengthen his squad before the transfer window closes on Friday.
He said: "I don't think it is my strongest squad (ever). We have just not got any injuries at the moment.
"I've got a bit of cover in some places but the numbers are down. But the flexibility we've got in the squad with [Steven] Naismith, Fellaini, [Leon] Osman, [Kevin] Mirallas, means we can cover things.
"But if we lost any group of numbers, we'd be struggling.
"I'd like to add to the squad if we could just to make sure we've got enough bodies. If I can add a loan or something came up which we could manage, then I would look to do it."
Villa gave the appearance of a side which needs to be strengthened before the transfer window closes.
Paul Lambert did not disclose his plans but admitted: "[Owner] Randy Lerner, myself and [chief executive] Paul Faulkner will sit and see what's going to happen.
"If something comes available and I think it's worthwhile, then you can look at it."
Villa, who won only four home games last season, were always chasing the game after Everton's early strikes.
Lambert said: "I was really disappointed. You can't in any game of football give someone a couple of goals start, particularly a team like Everton.
"You can't do it. The only pleasing thing was the second half - we kept going, which was pleasing, but the horse has bolted.
"You could sense the fans' frustration. Absolutely, you're 3-0 down at home! You can sense it. It's my responsibility. I take the blame for it.
"But there's a chance to rectify it on Tuesday [against Tranmere in the Capital One Cup]."
To add to Villa's misery, defender Ciaran Clark was sent off for a professional foul on Jelavic after 58 minutes.
Lambert said: "I don't think it was a malicious challenge. If he's the last man then the referee has done what he's done. It wasn't a bad tackle really."