With just five minutes of extra-time left, the Belgium international slotted the ball into the back of the net to set up a third-round tie away at Craven Cottage against Fulham
The 25-year-old, who only entered the game in the 95th minute, managed to prevent a nervy shoot-out for the hosts on Wednesday night after they failed to put the game to bed in normal time, despite numerous opportunities.
The Premier League side saw their opponents, who have strangely never played a League Cup tie away from home, go ahead through Luke Freeman but Barcelona loanee Gerard Deulofeu grabbed an equaliser just before the break.
But the wasted opportunities, many of which fell to Arouna Kone who made his first competitive start for the club, left the sides having to decide the encounter in extra-time, with Fellaini finally landing the decisive blow.
|By Tom Marshall-Bailey
Marouane Fellaini was the hero for Everton this evening as his late goal spared his side's blushes as they struggled their way to a 2-1 win over League One outfit Stevenage in extra-time.
But what remains to be seen is whether this is the Belgian giant's final farewell to the Goodison faithful, as the rumours of a move to Manchester United heat up ahead of transfer deadline day.
Given the pre-match suggestions that Leighton Baines has asked to leave the club, the post-match press conference will uncertainly be dominated by questions on the wantaway duo, but Roberto Martinez will have even bigger concerns at how much hard work his team made of inferior opposition here.
Ross Barkley was the main highlight for a side that will certainly have to improve on this performance if they hold serious aspirations of progressing in the competition. Fulham wait in the third round for the Toffees.
After early pressure from the hosts, the Ivorian should have given his side the advantage as Deulofeu got to the by-line and cut the ball back, but the former Wigan frontman's shot could not evade the Stevenage defenders blocking the goal line.
Despite Everton's dominance, it was Stevenage who took the lead when Freeman found the back of the net against the run of play in the 36th minute after some impressive build-up play from his team-mates. The former Arsenal forward fired home after a clever pass from Jimmy Smith, who found the 21-year-old with a neat pass into the area.
And though the home crowd saw more disappointment when Kone skied over minutes before the break, Deulofeu was on hand to claw it back immediately before the half-time whistle. The Spain Under-21 international kept his calm to glide forward and curl a neat left-footed shot across goalkeeper Chris Day.
The youngster came out for the second half with the same vigour, knocking a perfect ball into Steven Naismith with the 26-year-old just unable to turn it in.
His efforts typified a renewed fervour in attack from the home side, who heaped pressure on their opponents. Leon Osman went close for the Toffes, but his powerful shot was blocked by Doughty in the 58th minute.
Deulofeu remained the star of the show, and he came close to hitting the winner in the 71st minute when his effort from outside the box was blocked by Jon Ashton.
In an otherwise lacklustre end to the 90 minutes, Stevenage substitute Greg Tansey went close to a stunning winner, but Robles tipped his long-range effort over as the game entered extra-time.
Martinez opted to bring midfielder Fellaini into the mix in Barkley’s place, but it was Kone who had the best chance of the first period, with the Ivory Coast international dithering on the ball after Deulofeu’s precise cross.
However, Stevenage could not keep Fellaini out, and he made no mistake to fire home just yards from the goal to clinch the tie with barely five minutes left on the clock to set up a third-round trip to Fulham.