Fellaini headed the Toffees, victors in eight of their previous 10 clashes with City, into a first-half lead but then conceded a controversial penalty to allow Carlos Tevez to level.
Everton felt Edin Dzeko had gone to ground too easily to win the award but Tevez made no mistake and City went on to dominate the second half.
The result preserved City's unbeaten start to the season in the Premier League and prolonged their run of home games without defeat to 37, a sequence stretching back to a loss to Everton two years ago.
It was also Everton's seventh draw in their last nine games.
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.
Joleon Lescott made an unexpected return to the side, to face his former club, in place of impressive youngster Matija Nastasic.
The England defender looked composed as he resumed his partnership with Vincent Kompany after more than a month out of the starting line-up.
Everton playmaker Leon Osman gave the City defence plenty to think about in the early stages without creating a clear-cut opening.
Everton goalkeeper Tim Howard had to pick himself up from a collision with Dzeko after coming to claim a cross from Maicon, who looked dangerous in attack but vulnerable in defence.
City threatened the Everton goal in the 20th minute when David Silva crossed and Dzeko headed wide.
Tevez controlled well in the area but was unable to shoot and Dzeko inadvertently took the full force of a Samir Nasri shot in the back of his head.
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.
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 Joe 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 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.
Whether or not Fellaini was unfairly penalised, he almost produced the perfect response in the final moments of the first half with a glancing header from a Steven Pienaar cross, but Hart saved.
That proved the final on-field action before the players left the field but Moyes continued to voice his frustration, presumably about the penalty decision, to the fourth official.
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 almost broke away after a miskick by Distin but the defender recovered quickly to nick the ball away from the striker.
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.
City decided to freshen up their attack by sending on Sergio Aguero, but the crowd were not happy that it was Tevez he replaced and boos rang around the ground.
The game became scrappy as Everton's hard-working defence did their best to repel City's threat.
Mario Balotelli also entered the action for the last 10 minutes in place of Dzeko and did inject a bit of energy into the side.
His first chance to aim at goal, however, was probably best forgotten as he screwed a shot well wide.
Nasri then combined well with Silva in the area but could only win a corner.
Everton went close in injury time from a Jelavic free-kick but Hart saved and the ball squirmed wide.