Meulensteen's men, who ended a six-game Premier League losing streak by beating Aston Villa last week, reacted superbly after being overrun in the first half and deservedly equalised Leon Osman's early opener with a Dimitar Berbatov penalty midway through the second period.
However, the Londoners were level for just six minutes as Seamus Coleman put the hosts back ahead after 73 minutes, before Gareth Barry and substitute Kevin Mirallas put gloss on the scoreline with late goals.
The result - which extends Everton's unbeaten home run in the Premier League to 17 games - moves Roberto Martinez's side up to fourth, while Fulham remain in the relegation zone.
Everton controlled the early exchanges and their first real opportunity came when Phil Jagielka glanced a Steven Pienaar corner wide.
However, Martinez's men did not have to wait long for their dominance to be rewarded and Osman broke the deadlock after 18 minutes following a slick passing move.
Bryan Oviedo and Pienaar combined well down the left before the South African fed Osman on the edge of the area.
The 32-year-old first shaped to shoot with his right foot, before cutting back onto his left and bending a cultured effort past Maarten Stekelenburg to mark his 300th top-flight appearance in style.
Everton were buoyant after going ahead, with Gerard Deulofeu becoming increasingly influential down the right.
The Barcelona loanee - rewarded with a start after coming off the bench to score Everton's equaliser at Arsenal last week - beat John Arne Riise with ease before firing a low delivery across the six-yard box that just eluded Romelu Lukaku in the 21st minute.
Lukaku then went even closer to doubling the home side's advantage when Barry's header deflected off him from close range, only for Stekelenburg to pull off a superb reaction save.
Everton continued to probe for a second before the break, with Stekelenburg again called into action just after the half-hour mark to deny Coleman.
Martinez's side failed to maintain their tempo and fluency immediately after the interval, though, and were fortunate to stay in front when Alex Kacaniklic spurned a golden chance four minutes into the second half.
Tim Howard could only palm Steve Sidwell's long-range effort straight at Kacaniklic, but the Sweden international somehow contrived to shoot over from six yards with the goal at his mercy.
Fulham's confidence was visibly boosted by Everton's low-key start to the second period, and Berbatov deservedly brought them level from the spot after 66 minutes shortly after Deulofeu was withdrawn due to a hamstring injury.
Referee Anthony Taylor awarded the penalty after Barry had clumsily collided with Kacaniklic inside the area, and Berbatov stepped up to coolly send Howard the wrong way.
Despite the visitors looking the more likely to go ahead, Everton's first slick passing move of the second period resulted in Coleman restoring their advantage.
Sharp interplay between Pienaar and Lukaku down the left ended with Stekelenburg palming the South African's cross straight into the path of Coleman, who had the easy task of tapping in his third goal of the season.
And Everton sealed the points five minutes from time when Barry nudged home from two yards after Fulham failed to deal with a corner, before Mirallas - who came on for Deulofeu - got in on the act in injury time.
