Belgian international striker Lukaku, a constant menace to the Swiss champions on his competition debut, doubled United's lead with a textbook header soon after halftime.
Lukaku, who now has six goals in six games for United since joining from Everton, ran riot for the hosts, having an early goal-bound shot blocked by the chest of Manuel Akanji while Henrikh Mkhitarayan wasted two glorious chances.
First he headed over from in front of goal then somehow contrived to hit the post from Lukaku's low cross.
United lost Paul Pogba, captain for the night, with a hamstring injury midway through the half but it was his replacement Belgian Fellaini who put the hosts ahead when he headed home Ashley Young's inviting cross.
It was also Fellaini's first Champions League goal.
Winger Young was making his first appearance since last year's Europa League semifinal, second leg against Celta Vigo when he suffered a hamstring injury.
United doubled their lead eight minutes after the break in simple fashion. Daley Blind swung in a cross from the left and Lukaku climbed to head powerfully home.
Swiss champions Basle did cause United's defence some anxious moments and keeper David de Gea was alert to palm away a snap shot from Mohamed Elyounoussi.
United's level dropped in the second half but they finished strongly and Rashford made absolutely certain of the points when he stroked in the third goal after the hard-working Fellaini's deflected cut-back fell perfectly for him.
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