Chicharito capped a swift United breakaway to open the scoring in the sixth minute, only for the visitors to hit back and level before the break through Aras Ozbiliz's 20-yard effort.
The visitors went for broke after the interval, and had United on the ropes for a 10-minute period in which David De Gea produced a wonderful, point-blank save to deny Siem de Jong.
Nerves were set jangling with three minutes remaining when the unmarked Toby Alderweireld powered home a header from Ozbiliz's free-kick, but the Reds withstood the visitors' late pressure to squeak into the next round, where Athletic Bilbao await.
United's starting line-up featured four changes to that which had won in Amsterdam a week earlier, as Rafael Chris Smalling, Ji-sung Park and Dimitar Berbaov replaced Rio Ferdinand, Jonny Evans, Michael Carrick and Wayne Rooney respectively. Ferdinand's absence meant Phil Jones moved into central defence, while Ji-sung Park skippered the side.
Ajax, with a two-goal deficit to overturn, reprised their 4-3-3 approach with two changes from the side beaten a week earlier; as Nicolas Lodeiro and Ricardo van Rhijn stepped in for Ismail Aissati and the injured Dmitri Bulykin.
United sought to swiftly put the tie to bed and threatened within the opening minute. Berbatov held up the ball on the left flank and fed Nani, whose shimmy inside and low shot worked visiting goalkeeper, Kenneth Vermeer. Fortunately for the Ajax stopper, the ball rebounded just out of the reach of