The Red Devils were second best throughout the second half, following a dominant showing in the opening 45 minutes, and they came unstuck late on through a Michael Lang goal.
A point would have been enough to send Jose Mourinho's men through to the knockout stages as Group A winners, but they now require a point in their final match against CSKA in two weeks - five days before facing Manchester City in a crucial Premier League fixture.
Fellaini, handed his first start since September in one of seven changes from last time out, then diverted another header wide of target as the Red Devils closed in on a breakthrough goal.
For all their possession, however, United did not create another big chance until late in the opening 45 minutes when two arrived in quick succession.
Anthony Martial's delightful cross was glanced goalwards by Fellaini, who was denied by the outside of the post on this occasion, before Marcos Rojo's long-range drive came back off the crossbar on his first appearance since April.
Basel, needing a victory to keep qualification hopes in their own hands, looked much brighter early in the second period as Serey Die blasted narrowly wide and Mohamed Elyounoussi picked out the gloves of Sergio Romero.
Renato Steffen was the next FCB player to be thwarted by back-up United stopper Romero, who barely had to break a stride to keep out the latest shot from range.
In a complete contrast to the opening 45 minutes, it was Basel's turn to dominate possession as Steffen tried his luck with another couple of attempts, one of which deflected off Dimitri Oberlin and popped into the hands of Romero.
Mourinho turned to his bench with his side now desperately holding on for the point they were after, with Nemanja Matic and Marcus Rashford introduced, but it did not have the desired effect as eventual matchwinner Lang headed against the bar moments later.
Basel felt that a penalty should have been awarded 20 minutes from time after Chris Smalling stuck out a leg and tripped Steffen which, after a fair bit of deliberating, referee Daniele Orsato decided was not worthy of a spot kick.
That decision would not have mattered had Rojo not produced a sublime block to deny Oberlin in front of goal, while Serey Die had another effort helped over the bar by Romero as the final 10 minutes approached.
Not only did a 15th win in 19 outings this term elude United, though, they were also denied a share of the spoils in the final minute of normal time as Lang tapped home a cross at the back post to spark jubilant scenes around St Jakob-Park.
Basel, United and CSKA now head into the final round of fixtures knowing that just two of them can progress, with the Red Devils - three points better off - still the strong favourites to finish in top spot.