United were on the verge of failing to score for a third consecutive European home game until the Belgian turned and shot into the far corner, provoking Jose Mourinho to throw a box of water bottles to the ground in a mixture of joy and relief.
In keeping with United's struggles under Mourinho, it could have been even worse but for the brilliance of goalkeeper David de Gea who produced a stunning save from Ulisses Garcia's deflected effort to deny Young Boys midway through the second-half.
Mourinho made the bold decision to leave Paul Pogba and Romelu Lukaku on the bench and Alexis Sanchez out of his squad completely among six changes from a drab 0-0 draw with Crystal Palace on Saturday, but it largely backfired as United failed to impress once more.
However, the three-time European champions are guaranteed a place in the knockout rounds thanks to Fellaini's late intervention and Juventus' 1-0 win over Valencia.
Mourinho defended his decision to drop Pogba and Lukaku by referring to the clamour for Rashford to get more starts up front by certain sections of the English media.
But the Portuguese coach was less than impressed with the 21-year-old just five minutes in when Rashford raced onto Jesse Lingard's flick but dinked his finish over the bar with just goalkeeper David von Ballmoos to beat.
Brazilian midfielder Fred was handed a rare start in Pogba's absence despite his reported £50 million price tag when joining from Shakhtar Donetsk in June.
And Fred was inches away from his second United goal with a rasping drive that rose just over the crossbar in a bright start from the hosts that contrasted sharply to their sluggish form in recent weeks.
However, that initial flurry of excitement quickly died down as again Mourinho's men laboured to break down the Swiss champions, who they beat 3-0 in Bern back in September thanks to a Pogba double.
Djibril Sow backheeled an excellent chance wide just before half-time and Kevin Mbabu also fired into the side-netting as the visitors posed a threat on the counter-attack as United's search for a breakthrough became ever more desperate.
Rashford shot wide again from a narrow angle before Fellaini spooned over a glorious chance when Ballmoos flapped under pressure from Lingard.
Mourinho's patience finally ran out as he summoned Pogba and Lukaku from the bench for the final quarter.
But it was De Gea who came to the Red Devils' rescue at the other end moments later with a remarkable save to claw Garcia's deflected shot off the line.
Christian Fassnacht then blazed over another glorious opportunity to give the visitors a famous win.
And they were made to pay when Fellaini was fortunate to escape when he controlled Lukaku's flick-on with his hand before firing a shot into the far corner.