Head boy Fellaini came on as sub and then nodded United into the lead as they made a welcome return to the Champions League.
He showed just why boss Jose Mourinho rates him so highly after getting an early taste of the action because skipper Paul Pogba got crocked.
Fellaini struck in the 35th minute to finally give the Reds the reward their dominance deserved.
Ashley Young curled a beautiful ball across the face of the six-yard box and there was the mop-headed all purposes man to power it home.
Mourinho seemed to have pulled another master stroke after bulling up the big Belgian just hours before the game.
The goal seemed a long time coming after United missed a series of good chances but that didn’t seem to matter quite so much when Romelu Lukaku added a second in the 53rd minute.
He glanced a header home from close range after leaping powerfully to make sure he got to Daley Blind’s cross.
United certainly started brightly enough and were well in control of things when they were unexpectedly forced to make a change with only 18 minutes gone.
Skipper Pogba pulled up after stretching for a ball on the wet surface and it quickly became clear that he was going to take no further part.
On came Fellaini – the man who had been praised to the hilt by boss Jose Mourinho in the build up to the game.
Young was thrown the armband and United set about carrying on exactly where they had left off.
It seemed only a matter of time before they broke the deadlock after going at the Swiss champions from the word go.
They could easily have been in front after just 90 seconds when Romelu Lukaku thumped a shot goalwards only to see it ricochet out after striking the powerfully built Manuel Akanji.
While that was more of a speculative attempt from distance, Henrikh Mkhitaryan didn’t have the luxury of such an excuse when he missed a golden chance to give United a 12th minute lead.
He was perfectly placed to get a good contact with the ball when Juan Mata finished neat interplay with Lukaku with a deft chip towards the front post but his header didn’t even hit the target.
Midway through the half, Manchester United missed another golden opportunity to make the breakthrough and it was that man Mkhitaryan who was the culprit again.
Lukaku powered his way to the byline before ramming a low ball towards the near post where Mkhitaryan shot only to see it strike the post and when he got in a second effort, keeper Vaclik managed to smother it.
You just sensed that if United didn’t press home their advantage, they could get caught with a sucker punch and that was so nearly the case on the half hour when Luca Zuffi sent a powerful shot just wide.
Mourinho didn’t need to tell his players it was high time they punished the visitors – and in the 35th minute Fellaini answered the call.
That must have made for a calmer half-time team talk and when Lukaku popped up with the second any nerves were washed away in the torrential rain.
But they were given a sharp reminder not to take it easy when De Gea was forced to make a cracker of a flying save to keep out Mohammed Elyounouessi’s angled drive in the 63rd minute.
But United put the game beyond reach six minutes from time and it was that man Fellaini again in the thick of things.
He charged down the right before crossing and when Mkhitaryan missed the ball, there was sub Marcus Rashford to shoot home after drilling his shot into the ground and over the keeper.
MAN UNITED(4-2-3-1): De Gea; Young, Lindelof, Smalling, Blind; Pogba(Fellaini 18), Matic; Mata (Rashford 77), Mkhitaryan, Martial (Lingard 69); Lukaku. Subs: Romero, Carrick, Valencia, Darmian.BASEL(3-4-3): Vaclik; Akanji, Suchy, Balanta; Lang, Xhaka, Zuffi, Riveros (Oberlin 77); Steffen, van Wolfswinkel (Bua 66), Elyounoussi. Subs: Salvi, Gaber, Fransson, Schmid, Calia.REF: Ruddy Buquet (France).
