Danny Welbeck scored twice in two minutes in the first half for United, which collapsed after Fabian Frei pulled Basel's first goal back in the 58th minute.
Alexander Frei headed Basel level two minutes later after meeting Fabian Frei's cross and then put the Swiss champions in front from the penalty spot in the 76th after Antonio Valencia fouled Marco Streller.
But Young leveled with a downward header in the 89th after meeting with Nani's cross.
The draw leaves United winless and with just two points after two matches, while Basel is top of the group with four.
"We were careless - it's a wake-up call in many ways," United manager Alex Ferguson said. "A lack of concentration saw us give the game away really, but we rescued it."
In the first half, United did not appear to be missing its injured first-choice strike force of Wayne Rooney and Javier Hernandez as Welbeck delivered.
Ryan Giggs played a key supporting role to Welbeck and was devastatingly influential in providing the vital touches to twice allow the 20-year-old striker to finish.
The first came from a well-worked move in the 16th minute. Phil Jones brought the ball forward, Valencia took over and picked out Fabio da Silva's run into the box. A simple ball across the box was cushioned by Giggs into Welbeck's path and although he scuffed his shot slightly it still found the corner of the net.
Just over a minute later, Giggs played through a precise ball that Welbeck struck home with a powerful low shot to make it 2-0.
Strellar and both Freis missed the target from decent positions as Basel attempted to reduce the deficit, but the second half was a different story.
United goalkeeper David de Gea had saved with his foot from Streller before Fabian Frei put the visitors back in the game.
Streller headed on a corner and when the ball reached Fabian Frei 12 yards (meters) out he volleyed in off the upright.
Fabian Frei then turned provider, crossing for Alex Frei to leave de Gea grasping at thin air as his header soared into the net to equalize.
Winger Nani was thrown on in place of defender Fabio, with Valencia dropping to right back.
Nani's first contribution was a negative one, however, as he gave the ball away that led to the penalty kick.
The Red Devils' failure to clear their lines cost them dearly as Streller burst into the box and Valencia brought him down.
Alex Frei made no mistake from the spot, blasting his shot high past de Gea to put Basel in front.
"To concede three goals at home, the defense and midfield have to be better than that in terms of concentration," Ferguson said. "If you lapse, the quality of the Champions League can make you suffer."
Ferguson sent on Dimitar Berbatov with nine minutes left and the Bulgaria striker had one chance with a header but Yann Sommer made a comfortable save.
United continued to pile on the pressure, with Nani's cross finding Young at the far post and the England winger heading powerfully past Sommer.