Goals from Victor Anichebe, Kevin Mirallas and Marouane Fellaini gave the Toffees a comprehensive victory and extended their record of having never lost to Swansea.
The win also continued Everton's fine start to the season and moved them up into second in the table.
Starting a season well has proved to be a problem for the Merseysiders over recent campaigns, but hopes are high among the Everton faithful this time round, as exemplified by their chants of "We're going to win the league" during the dying embers of the contest.
Moyes stopped short of predicting a title challenge or Champions League qualification, but is confident his side can mix it with the sides at the top of the table.
"It was great to hear the fans singing that as we have not heard it often at Everton," he said.
"If we had won midweek against Newcastle as we should have done, we could have been mixing it with the big boys, which we are still doing and we want to be doing at the end of the season.
"We have finished a lot of seasons strongly and the one year we got off to a good start we qualified for the Champions League and we have to do that if we can.
"We will probably finish where most people would predict and we will try and do that and we will do better if we can."
Moyes also praised the performance of Anichebe, as the striker found the net for the second successive game and turned in a powerful display in place of the injured Nikica Jelavic.
Anichebe's pace and physicality was a constant thorn in the side of the Swansea defence and Moyes said: "He did okay in midweek against Newcastle as well and did not get the credit he deserved for his goal and the one that should have been. We are playing with confidence and he is contributing well.
"He does not possess an awful lot of self-confidence but if he gets that and has a bit of belief he is a handful for defenders as he has a lot of attributes that are difficult to deal with in the modern game."
Swansea played a part in their own downfall with a sloppy opening half-hour from which they were fortunate to emerge just a goal down.
Having found a foothold in the game they were ruthlessly hit on the counter for Mirallas' goal, as Everton strung together a superb move which started in their own right-hand corner with Phil Neville, Steven Pienaar and Fellaini combining brilliantly.
Angel Rangel and Ki Sung-yueng went close to pulling a goal back but Nathan Dyer's dismissal for a second yellow card just 12 minutes after coming on at half-time effectively ended their hopes.
But Laudrup made no excuses, declaring the display the worst since he took over.
He said: "I have to say Everton deserved the win, they were better than us.
"We were disappointing in the first 30 minutes and we gave away a lot of fouls, they had five or six free-kicks in the first 10 minutes and we played too slow and the first 30 minutes laid the ground work for the defeat.
"We can learn from this. We were 2-0 down and had a good reaction, but the first 30 minutes was very disappointing, the worst we have been this season until now."
Dyer had already been booked for dissent when he clumsily caught Leighton Baines. Laudrup had no issues with the decision but felt the winger had been "unlucky".
He said: "The second yellow was a little unlucky, Nathan went to kick the ball, he missed the ball and in the same movement kicked his opponent but it was a yellow card and you can't say anything about it."
Dyer had replaced record signing Pablo Hernandez at the break, and Laudrup confirmed the Spaniard was withdrawn due to a knee problem.
He said: "He had small problem with his knee, he says it's a little on the side of his knee but we don't need more injuries."