Marouane Fellaini, who came on for the injured Paul Pogba in the first half, rose highest to head United ahead from an Ashley Young cross with 10 minutes to go to the break.
Romelu Lukaku strengthened their grip on the encounter by converting from among a group of players in the box from a Daley Blind cross following a short corner.
Marcus Rashford fired in his side’s third from a Fellaini low cross seven minutes after coming on. The striker has now scored in his first appearance in the Champions League, Premier League, Europa League, FA Cup, League Cup and for England.
The scoreline could have been even greater for Jose Mourinho’s dominant side, making their first appearance in Europe’s premier club competition since 2015, having qualified by winning the Europa League in May.
They created numerous clear-cut chances, the best of which Henrikh Mkhitaryan side-footed against the post from close range after being found by a Lukaku cut-back.
Basel were pinned back in their own half for much of the game but could have forced a different outcome had Luca Zuffi not fired wide from an unmarked position in the box just before Fellaini’s opener.
Mohamed Elyounoussi also forced David De Gea into a fine save with a rising shot from inside the box as the visitors became more ambitious in the final quarter of the match.