Fellaini, a replacement for the injured Paul Pogba, gave United a 35th-minute lead at a drizzly Old Trafford yesterday and teed up Marcus Rashford for the hosts' third goal after Romelu Lukaku had struck early in the second period.
It was United's first Champions League game since December 2015, with Jose Mourinho's side having engineered a return to the competition by winning the Europa League last season.
Basel had previously proved a thorn in United's side, eliminating them from the Champions League in 2011-12, and they inflicted a surprise defeat upon Mourinho in the first European game of his second Chelsea stint in 2013.
With Ashley Young coming into the United XI at right-back in place of Antonio Valencia to make his first start of the season, Pogba inherited the captain's armband from the Ecuadorian.
But the Frenchman's debut as United skipper would last just 18 minutes after he abruptly clutched the back of his left leg and was obliged to hobble off in disgust, with Fellaini joining the fray.
Luca Zuffi gave the hosts a scare when he sliced wide in front of the Stretford End after Chris Smalling failed to deal with a high ball, but two minutes later United opened the scoring.
- Lukaku opens his account -
============================
The home fans could have been forgiven for wondering if it was still 2015, when the stadium last hosted a Champions League game, as Young sidestepped Blas Riveros and whipped in a perfect cross that Fellaini headed home.
The resulting corner, taken short, yielded United's second goal as Daley Blind's drilled cross was emphatically headed in at the back post by Lukaku, who notched his first goal in the tournament proper.
Frequently maligned for his ungainliness, Fellaini by now was running the show, freed to roam forward by the anchoring presence of Nemanja Matic behind him.
Martial should have made it 3-0 from the Belgium midfielder's cut-back midway through the second half, only to steer his shot off-target from close range.
Mkhitaryan completely miscued his attempt at a side-foot finish from Fellaini's right-sided cross, but substitute Rashford arrived behind his team-mate to scuff a shot into the ground and over the despairing Vaclik.
Off the pace domestically, Swiss champions Basel have now gone nine European games without a win.
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