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.
Kenwyne Jones grabbed a hat-trick for Stoke as they beat Walsall 3-1 and Villa wrapped up a 3-0 win over Rotherham with goals from Andreas Weimann, Christian Benteke and Fabian Delph.
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.
(Reporting by Toby Davis; editing by Ed Osmond)