The Scot claims UEFA Champions League qualification has always been in the Toffees' thoughts, with a consistent campaign having kept them within striking distance.
They are now just four points off the pace with a game in hand and will play host to fourth-placed Arsenal at Goodison Park next weekend.
"We have believed all along that we can compete for the Champions League places, and that's what we will continue to do," Naismith told Sky Sports.
"This win here is just as big as next week is going to be now. I'm sure the manager will have his game plan for it and we'll work on it this week and hopefully get another win."
On Everton's showing at Craven Cottage, which saw them grind out a fifth successive win against opponents scrapping for their Premier League lives, Naismith added: "It was a big win, especially at this stage of the season.
"You probably start taking results over performances and today we were probably a bit lethargic in the first half, but it shows the quality we have got amongst us that we stuck in and by the end were controlling the game."
Naismith was introduced at the interval in place of Ross Barkley - after he picked up a knock - and the former Rangers forward was influential - along with his fellow substitutes - as he forced an own goal from David Stockdale and wrapped up the points after Kevin Mirallas had added a second.
Asked what Roberto Martinez had asked for at half-time, Naismith said: "He just wanted it to be a bit more lively going forward. There was a lot of space to try and make things happen, and that's what I tried to do.
"For the first goal, it dropped to me at the edge of the box and I'm just trying to get it on target, and then the last one for me was to seal the win and take the pressure off at the end of the game."
Naismith was relieved to bundle home Everton's third of the afternoon as his celebrations for the opener were cut short by the fact that a hopeful drive through a crowd of bodies took two deflections on its way into the back of the net and will be credited to Fulham keeper Stockdale.
He added: "If I hadn't scored the third I would have been complaining about that! But overall, the main thing is the result. At this stage of the season you are pushing for Europe and the wins are vital."