The Belgian, at the centre of a transfer tussle with Manchester United, bundled in from close range after a goal-mouth scramble with 115 minutes on the clock.
A day after their Merseyside neighbours Liverpool were also taken to extra-time by third-tier Notts County, Everton found themselves in a similar spot of bother when they fell behind to a first-half Luke Freeman goal.
They drew level when Spanish teenager Gerard Deulofeu, on loan from Barcelona, netted following a mazy run on the stroke of halftime and Fellaini's goal set up a third-round clash with Fulham.
Newcastle United were held for 80 minutes by fourth tier Morecambe before substitute Shola Ameobi found the net with a deflected shot and his younger brother Sammy completed a 2-0 win with a calm finish.
There were no such difficulties for Aston Villa and Stoke City who both scored three times to move into the next round.
Premier League Cardiff City were pushed hard by fourth-tier Accrington Stanley but broke through in the 61st minute when Nicky Maynard finished after a neat flick and Rudy Gestede doubled the lead a minute later to complete a 2-0 win.
Championship leaders Nottingham Forest needed extra-time to beat fellow second-tier side Millwall 2-1 with Jamaal Lascelles heading the winner after 94 minutes.
Watford sealed their place in the third round with a 2-0 win over Bournemouth thanks to an Elliott Ward own goal and a chipped finish from Cristian Battocchio.
- Sports & Recreation
- Manchester United
- Gerard Deulofeu