Coleman ignited a hitherto drab Premier League affair at Goodison with a well-placed header 10 minutes before the break.
Former Fulham striker Saha doubled the Toffees’ lead from a free-kick early in the second half but a Clint Dempsey strike prevented the home side easing to victory.
The game began at the pace of a typical end-of-season mid-table encounter, despite both teams having another eight games remaining.
The spirit Fulham showed in the second half made their lack of urgency in the first seem puzzling but the Londoners never looked like winning at Goodison for the first time in 23 league visits.
Defeat could have been heavier had Carlos Salcido not produced a fine headed clearance off the line from Phil Jagielka late on.
Everton were always the dominant side and fully deserved to repeat their victory over the Cottagers in Moyes’ first match just over nine years ago.
The downside was the loss of Saha, who was playing for Fulham in 2002, on a stretcher in the closing minutes.
The Toffees had welcomed back captain Phil Neville and the influential Tim Cahill from injury.
Neville played in midfield in the absence of Mikel Arteta while Cahill was pushed up front with Saha. Tony Hibbert, a survivor from Moyes’ opening game, was at right-back.
Fulham were unchanged from their last outing against Blackburn a fortnight ago after Dickson Etuhu overcame a hamstring strain.
The game lacked bite early on but Everton had the better of the play, with Leon Osman curling an effort over and Leighton Baines dragging a shot across goal.
Fulham had an opportunity when Neville bundled former Everton striker Andy Johnson over on the edge of the box but the free-kick came to nothing.
Saha threatened with a run up the middle but Brede Hangeland denied him with a fine tackle.
Saha had another opportunity when he rose to meet a Baines corner but his header bounced and Salcido calmly headed off the line.
Baines then delivered a fine cross for Jack Rodwell but the midfielder’s header did not test Mark Schwarzer.
Everton made the breakthrough after Cahill had a shot blocked and the ball rolled out to Baines on the left.
Baines cut it back inside for Osman and the midfielder lifted a cross to the back the post for Coleman to place a firm header back across goal and beyond Schwarzer.
The game at last had the spark it needed and moments later Coleman was stopped in full flow on the edge of the area but the resulting free-kick from Baines was blocked.
Fulham had to wait until first-half injury time for their first serious opportunity as Dempsey shot narrowly wide after good work by Johnson.
Everton took a firm grip on the game four minutes into the second half after Rodwell was brought down just outside the area by Etuhu.
Baines tapped the free-kick to Saha and the Frenchman blasted a low free-kick through the wall for his 10th goal of the season.
Fulham tried to respond but Etuhu’s shot into a crowded area was blocked and Damien Duff fired the rebound straight at Tim Howard.
The tempo was much quicker than in the first half and Coleman forced a good save from Schwarzer with a strong run and shot from distance.
Fulham boss Mark Hughes decided to make a change and was instantly rewarded after sending on Bobby Zamora, still being eased back into action after a broken leg, for Etuhu just before the hour.
Zamora, with his back to goal, brilliantly laid off a cross from Duff and the inrushing Dempsey struck a powerful low shot into the bottom corner from the edge of the area.
Jagielka almost made the game safe for Everton with an inch-perfect header but Salcido did brilliantly to again head away from just inside the post.
Zamora failed to make the most of a headed chance at the other end after being picked out by Gael Kakuta.
Saha’s afternoon ended in agony in the closing minutes after he landing awkwardly following an aerial challenge.
He was replaced by Jermaine Beckford, who spurned a good chance to add a third on the break in injury time.