United were dominant for much of the game but were hit by a late Michael Lang sucker punch and made to wait for qualification.
The Reds are still in a strong position to seal their place in the last 16, and only need a point from the final match at home to CSKA Moscow to go through.
Indeed, the only way United cannot qualify for the knockouts would be if Basel beat Benfica and we lost by a margin of six goals or more - a very, very unlikely scenario.
Sergio Romero and the much-changed Reds had held firm until the final moments when Raoul Petretta's cross evaded everyone and Lang nipped in to score from close range.
United, who welcome back Marcos Rojo at the heart of the defence, had been in control virtually from the off, with the hosts restricted to the odd effort from distance.
Captain for the night Paul Pogba was again commanding, and it was from his intricate build-up play that the Reds carved out the first opening.
The Frenchman's pinpoint through pass picked out Romelu Lukaku on 12 minutes, but Tomas Vaclik was out smartly and saved well.
Marouane Fellaini headed wide, Pogba went close with a set-piece and Dimitri Oberlin's free-kick drifted a long way off target at the other end.
The Reds should have taken the lead shor
Rojo cleared well from Renato Steffen's dangerous inswinger, Lukaku headed wide from a Martial centre, before Serey Die thumped wide from distance as Basel started to up the ante.
Vaclik saved superbly from Martial before United almost moved ahead through the boot of an unlikely source, when Rojo almost marked his comeback with a goal after seven months out.
The Czech keeper had been impressive for the shots but this time he could only watch on as the Argentinian's deflected 30 yarder left him grasping at thin air, only for the crossbar to come to Basel's rescue.
Tricky winger Steffen had a shot parried by the recalled Romero, then he cut inside Daley Blind and bent an effort narrowly wide.
At the other end, Pogba had a well-struck effort held by Vaclik and Lang came within inches of the elusive breakthrough when he powered against the bar from a corner.
Basel had a penalty appeal waved away, Oberlin went close and substitute Marcus Rashford fired into the side netting on the angle as the tie opened up in search of a winner.
Knowing they had to win after CSKA's victory over Benfica in the other game, Basel continued to press and came on strong in the closing stages.
Die forced a flying save from Romero, while substitute Zlatan Ibrahimovic headed over and then blasted a speculative effort wide.
Then came Lang's late sting in the tail as United were beaten in Basel again, but, this time the qualification picture still looks pretty for the Reds.
United Faithful Man of the Match: Paul Pogba
Overall team performance: 6/10