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.
"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."
He added: "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."
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.
Moyes 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."
Laudrup made no excuses after the defeat, declaring the display the worst since he took over.
"I have to say Everton deserved the win, they were better than us," he said. "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."
Source: PA
Source: PA