Missing Wayne Rooney due to a throat infection, United squeezed through 3-2 on aggregate after a nerve-wracking finale and another shaky performance at Old Trafford where they have won one of only four European matches this season.
United, 2-0 ahead from the first leg, appeared to be cruising when Javier Hernandez put them ahead after six minutes with a clinical goal.
The slick-passing Dutch side refused to be overawed and leveled with a superb left-foot strike from Aras Ozbiliz in the 37th minute despite a suspected handball by Nicolas Lodeiro in the buildup.
Ajax, the first Dutch team to win at Old Trafford, took control of the game and were unlucky not to go ahead just before the hour when Siem de Jong's header was brilliantly turned over the goal by David De Gea.
Although United eased Ajax's grip after Paul Scholes came on, defender Toby Aldeweireld stole ahead of hesistant defenders to head home from an inswinging free kick in the 87th minute.
