On 17 minutes Darren Bent timed his run to perfection to meet Kenwyne Jones’ cross with a firm downward header which beat Everton keeper Tim Howard.
But the visitors, who had earlier seen Leon Osman clip the outside of the post with a sixth-minute effort and Louis Saha twice fire wide from promising positions, responded well.
On 54 minutes Saha volleyed just wide.
Howard truned over Paulo da Silva’s powerful header from a 68th-minute Andy Reid free-kick before Steven Pienaar whistled a long-range effort just wide and Saha pulled another shot across the face of goal.
But with five minutes remaining Fellaini controlled Tony Hibbert’s cross before smashing a shot high into the net.