Influential Toffees midfielder Fellaini drew first blood for the visitors on the 33rd minute at the Etihad Stadium as he saw his initial headed effort from close range parried by City goalkeeper Joe Hart before bundling home the rebound.
But Fellaini went from hero to villain 10 minutes later when referee Lee Probert pointed to the spot after he appeared to tug Edin Dzeko in the penalty box and Tevez made no mistake slotting home past Tim Howard down the middle.
The match took some while to warm up and City suffered an early blow as Aleksandar Kolarov limped off after just six minutes. Manager Roberto Mancini had opted to rest the hard-working Pablo Zabaleta but the in-form Argentinian was summoned from the bench straight away.
Everton's 33rd-minute opener came in familiar fashion as the ever-dangerous Leighton Baines, who passed a fitness test to feature, crossed and Fellaini eventually headed in.
Vincent Kompany did well to divert the cross away from Nikica Jelavic but it dropped for Fellaini, who headed home at the second attempt at the far post after Hart saved his initial effort. It was the Belgian's eighth goal of an impressive season.
That brought the game to life and City almost found an immediate equaliser as Tevez met a Samir Nasri cross with a deft header that Howard did well to save.
Gareth Barry then chested down another Nasri cross for Dzeko to turn and shoot but again Howard was alert.
Controversy followed the resulting corner as Dzeko tangled with Fellaini and went to ground. Referee Lee Probert sparked immediate protest from Everton by pointing to the spot and Toffees manager David Moyes clearly thought the award was harsh.
Tevez made no mistake as he stepped up to tuck home his eighth of the campaign.
City upped the tempo at the start of the second half but again it took time before the chances came, aside from a Tevez shot which was blocked by Phil Jagielka.
Tevez also headed over but it was not until just after the hour that City seriously threatened again as Maicon burst down the field and forced Howard to parry a powerful 18-yard drive.
The game became scrappy as Everton's hard-working defence did their best to repel City's threat, then the Toffees went close in injury time from a Jelavic free-kick but Hart saved and the ball squirmed wide.
|Manchester City||Team Statistics||Everton|
|1||1st Half Goals||1|
|5||Shots on Target||4|
|2||Shots off Target||3|
|50.1||1st Half Poss.||49.9|
|68.3||2nd Half Poss.||31.7|