Young steered in a 90th-minute header from a Nani cross to deny Basel a famous victory after two goals from Alex Frei and one from namesake Fabian Frei had fired the Swiss side into a 3-2 lead at Old Trafford.
United, missing the injured Wayne Rooney, had looked to be cruising to victory early on after two first-half goals in the space of a minute from Danny Welbeck helped them into a 2-0 lead at the break.
But Sir Alex Ferguson's men were left shellshocked by a superb second-half fightback from the Swiss visitors, who ruthlessly punished slack defending by the English champions to go within a whisker of a famous upset.
"We were careless. They made two or three chances in the first half but a lack of concentration killed us," Ferguson said afterwards.
"It was a really good performance in the first half but we were left grappling to get a result in the second half. We threw it away.
"Losing three goals at home should be a wake-up call for us."
The draw leaves Basel on top of Group C with four points from two games, with Portuguese side Benfica second on goal difference. United are two points off the pace after two games.
United were given a let off after only six minutes, Basel captain Marco Streller sending Fabian Frei racing into the box with a deft reverse pass through the legs of defender Phil Jones.
The covering Rio Ferdinand did just enough to unsettle Frei but the striker should have done better with his shot, which flew high and wide.
United's response was swift, Ashley Young beating his marker before curling a cross to the far post which Antonio Valencia headed wide.
Basel were back under pressure soon afterwards, goalkeeper Yann Sommer forced to save from Young's shot on 14 minutes after more good work from Valencia.
With United probing Basel menacingly, it appeared only a matter of time before the visitors defence buckled and on 16 minutes United took the lead.
Valencia sent fullback Fabio charging into the box with a perfectly weighted pass that sprang the Basel offside trap, and the young Brazilian squared for Ryan Giggs who laid off for Welbeck to screw a low angled shot in off the post.
The celebrations from that effort had only just subsided when Welbeck grabbed his second a minute later, Giggs threading an exquisite pass through to Welbeck for the United striker to stab home.
Yet the remainder of the half was anything but one-sided, and had Basel possessed a more clinical finisher the visitors might easily have scored, Alex Frei squandering good openings in the 24th and 36th minutes.
United maintained their commanding two-goal lead at half-time but Basel were far from ready to concede defeat after the break and should have pulled a goal back on 51 minutes through Streller.
Once again, United's defence were caught out by lax marking and Streller found himself with only David de Gea to beat.
But with the goal at his mercy Streller took a fraction too long to get his shot off and the Spanish keeper was able to block with his legs.
That ought to have been warning enough for United, but seven minutes later the English champions were found wanting again.
De Gea saved brilliantly from a Streller header but United were slow to clear and Fabian Frei had time to lash home for 2-1.
Two minutes later and Basel were level. Granit Xhaka sent Fabian Frei racing away down the right flank with a lovely ball, who then floated a wonderful cross into Alex Frei who steered his header beyond de Gea to level it.
On 75 minutes United's worst fears were realised when Jones tried to pass the ball out of defence and Basel went on the attack once more.
Streller darted forward onto the loose ball and when Valencia bundled him over, referee Paolo Tagliavento immediately pointed to the spot.
The veteran Frei duly stepped up to shoot Basel ahead. That looked to have settled it until relentless late pressure yielded Young's late strike.