The visitors ended a 12-game Premier League run without a win and rekindled their hopes of avoiding relegation.
Fulham is now in 18th spot on 18 points from 23 games, seven behind 17th-placed Newcastle United in the safety zone, while Everton stayed seventh on 37 from 22 after another lacklustre home performance.
They suffered a third home defeat in four games, while Fulham won at Goodison Park for the first time in the league after 22 straight defeats, with Maja netting on his first start for the Cottagers.
Maja, 22, was upbeat after the longest home winning run over an opponent in English top-flight history ended for Everton, who had beaten Fulham repeatedly since 1961.
“It is a big one. We came into the game wanting to win it and had not had the results we wanted,” he told BT Sport.
“We took our chances when they came and put in another good performance again. We want to continue like this.
“For the whole 90 minutes the guys worked hard and stuck to the game plan.
“This is a night I have been dreaming about for a long time, especially last night, and I had a feeling it was coming and am pleased it became a reality.
Sunderland striker, on loan from French side Girondins Bordeaux, slid in to steer home Ola Aina’s fizzing low cross in the 48th minute.
He made it 2-0 in the 65th with another predator’s finish, popping up in front of goalkeeper Robin Olsen to tap in a rebound from close range after Harrison Reed hit the post with a long-range effort.
Everton striker Josh King had a goal disallowed for offside in the closing stages, which summed up the Toffees’ miserable evening as they failed to create anything meaningful up front.
Fulham’s former Everton forward Ademola Lookman was delighted.
“We kept believing,” he said. “At halftime we said we’ll do the exact same thing in the second half and we scored twice.”
“It gives the team a boost of confidence and we’ll take that into the next game.
“I’ve said before if we stick together and put the ball in the back of the net, we’ll win games.”
