They were always up against it at St-Jakob Park once Marco Streller had given the hosts a ninth-minute lead.
Though they grew in dominance as the match went on, the elusive equaliser would not come.
Wayne Rooney, who faces a UEFA disciplinary hearing in nearby Nyon on Thursday, wasted their best chance, with Markus Steinhofer belting the ball against his own crossbar midway through the second half.
Alexander Frei added Basel's second eight minutes from time, although United did not go down without a fight.
Phil Jones grabbed a lifetime near the end but the visitors were unable to conjure up a miracle second, leaving Basel to celebrate the greatest night in their history.
In stark contrast, United are counting the massive financial implications of exiting at this stage for the first time since 2005, the lost millions hitting the Glazer family where it hurts the most.
On that occasion, United at least avoided the humiliation of appearing in the Europa League.
Not this time though. After the catastrophe of their 300th European game, United's 301st will be played on a Thursday night.
By half-time, a night that started with the prospect of having to face a fellow European heavyweight in the last 16 was turning towards the prospect of a meeting with Vaslui, Legia Warsaw or Maccabi Haifa.
Although they had started well, United suffered an early hammer blow Ferguson cannot have expected.
A penalty area collision between Nemanja Vidic and Chris Smalling offered Xherdan Shaqiri the opportunity to collect a loose ball and drive a long way down the by-line before drilling a cross across the visitors' goal.
It was not an easy moment for David de Gea. But the young Spaniard might have chosen a better option than to lunge at the ball with his feet, succeeding only in turning it to Streller, whose first-time shot bounced in.
For a good while afterwards United struggled to find an answer to Basel's blanket defence, an ominous sign in itself given the essential need for a goal.
Nani was their best hope, daring opposition defenders to lunge in before escaping their presence with a burst of acceleration.
He must have thought it had done the trick when he sent a low cross to the far post that appeared to set Rooney up with a tap-in at the far-post. Inexplicably, the striker failed to make proper contact.
Park Ji-sung could not react quickly enough either and Yann Sommer snuffed out the danger.
Sommer denied Nani and Ashley Young as the pressure began to mount, with Phil Jones having a shot blocked too.
But the second shattering moment of the half came in the final minute when Vidic and Streller chased down possession, got their legs tangled up and hit the deck.
The Serbian immediately called for assistance and medical staff wasted no time stretchering him down the tunnel, triggering the introduction of Jonny Evans and much half-time soul-searching within the visitors' ranks.
It was the type of situation Roy Keane excelled in.
Sadly for United, the Irishman's presence was limited to that of touchline TV analyst.
Rooney was presented with yet another opportunity four minutes after the restart when Giggs sent him clean through.
The angle was not great but, after opening his body to make room for the curling shot to the far corner, the effort sailed wide.
Even for a team of United's stature, hauling themselves back from two goals down would have been virtually impossible, so De Gea's brilliant fingertip save to turn over Alexander Frei's goalbound free-kick was priceless.
And the visitors so nearly benefited from an outrageous slice of good fortune when, in attempting to hack Nani's cross clear at the far post, Steinhofer succeeded only in rattling the crossbar.
It was one of those moments when fans start to wonder whether this is going to be their night. Rooney heading Nani's cross over from close range was another although, as the striker was offside anyway, it did not matter.
United plugged away for a bit longer but it seemed the life had gone out of their challenge as Frei finished from close range.
Typically the visitors rallied, scoring through Jones, but it was not enough.