The Blues easily extended their unbeaten run to seven games as a master-class by David Moyes' men moved them closer to the European places.
Clear favourites for three points, it took the home side only 12 minutes to take the lead through Yakubu.
Route one from goalkeeper Tim Howard had the Sunderland defence in chaos as the centre-forward latched on to the ball and hit a deflected shot over stranded Craig Gordon.
The Blues' lightning start continued as they doubled their lead five minutes later through free-scoring Tim Cahill.
Mikel Arteta found Phil Neville with a neat little flick and the England man crossed the ball for Yakubu to nod down into the path of Cahill who smashed his shot past the keeper.
Without a single win on their travels this season, Sunderland didn't look like breaking the habit and the game was all but won on 43 minutes as Steven Pienaar grabbed the third.
The South African winger linked up with left-back Nuno Valente and played a double one-two before the 25-year-old blasted the ball into the roof of the net.
After putting up little resistance for 45 minutes, Sunderland sprang into life just before the break and pulled one back through veteran striker Dwight Yorke.
Good play from the visitors forced a double save from Howard after shots from Grant Leadbitter and Carlos Edwards had the defence in a panic.
The ball rebounded to the unmarked striker and the former Manchester United front-man found the net one minute in to stoppage time.
The second half was no easier for the visitors as the Blues once again made the newly-promoted Black Cats look like a pub team.
A long searching ball from Joseph Yobo found Cahill one-on-one with the centre-half and, after a slip from Paul McShane, the Everton man slotted in his second on 62 minutes.
Record signing Yakubu had his best game in an Everton shirt to date and side-footed his second of the match on 73 minutes to make the score 5-1.
Coming off the bench from the restart Andy Johnson looked sharp and within six minutes the striker out-paced the defence to chip the ball over a committed Gordon.
Leon Osman inflicted more misery on the traveling fans by picking up the ball on the halfway line and running through the whole defence before sliding the ball in to the bottom corner for number seven.