Marouane Fellaini came off the bench to open the scoring in the first period, before Romelu Lukaku and Marcus Rashford netted after the break as the 20-time English champions secured three comfortable points on their Champions League return.
There was also a completely new back four, with Ashley Young, Victor Lindelof, Chris Smalling and Daley Blind all selected. Young came in for his first appearance of the season, while Lindelof was given his Old Trafford debut due to suspensions for Eric Bailly and Phil Jones.
As for Basel, former Norwich City striker Ricky van Wolfswinkel led the line for the visitors, while Taulant Xhaka, who is the brother of Arsenal midfielder Granit Xhaka, also started in a 3-4-3 formation.
Man United might have made the breakthrough inside the opening two minutes when a half-cleared corner fell to Lukaku inside the Basel box, but the Belgian saw his strike blocked, before Martial fired over at the far post.
The home side's first serious opportunity of the match arrived in the 13th minute when Mata clipped a super cross onto the head of Henrikh Mkhitaryan, but the Armenian missed the crossbar from a central position, before Lukaku stung the palms of Basel goalkeeper Tomas Vaclik moments later.
Paul Pogba was named Man United captain for the European tie, but his evening lasted just 19 minutes, with the Frenchman picking up a hamstring problem, which saw Fellaini introduced in the middle of the park.
Mkhitaryan had another super chance for the home side in the 22nd minute after meeting a low cross from Lukaku, but the attacker could only hit the post from close range as Basel survived another dangerous moment.
The away side had a golden opportunity of their own 10 minutes before the break when the ball fell kindly for Luca Zuffi just inside the Man United box, but the number seven fired his effort hopelessly wide of the post.
Mourinho's side finally made the breakthrough in the 35th minute when Fellaini rose highest to head a pinpoint Young cross into the back of the net, and it was no more than the Red Devils deserved after dominating the first period.
Basel had another half-chance in the latter stages of the first half after Mohamed Elyounoussi met a free kick from the right, but the attacker could only head into the arms of David de Gea, and 1-0 was how it stayed entering the half-time interval.
Man United doubled their lead in the 53rd minute, however, when Lukaku headed a super Blind cross into the top corner as the Belgian centre-forward registered his first ever goal in the Champions League.
Elyounoussi was Basel's biggest threat on the night, and the attacker sent another effort wide of the post just before the hour as the visitors continued to struggle in the final third of the field.
Lukaku had a wonderful chance to score his second in the 62nd minute, but Vaclik was on hand to keep out the striker's header, before De Gea made a smart save to deny the dangerous Elyounoussi down the other end of the field.
Martial was next to come close for the home side after meeting a low cross from Fellaini, but the Frenchman could not turn his effort towards goal, and moments later, Vaclik had to make a smart save to deny Young.
Rashford was introduced for his Champions League debut in the 77th minute, and the England international got his name on the scoresheet seven minutes later when he finished at the far post after excellent work from Fellaini.
Basel's supporters remained in fine voice in the latter stages of the contest, but the Swiss outfit struggled badly in Manchester as Mourinho's side opened their Group A campaign with a strong win.
Man United will now switch their attention back to the Premier League as they prepare to host Everton this weekend, while Basel have a Swiss Cup match away to Chiasso on Sunday afternoon.