United entered the game with a 2-0 lead from the first leg. It advanced 3-2 on aggregate.
United will face Atletic Bilbao in the round of 16 after the Spanish club beat Lokomotiv Moscow 1-0 to advance on away goals.
Also advancing were Valencia, Atletico Madrid, PSV Eindhoven and Udinese.
Javier Hernandez opened the scoring for United in the sixth minute and Aras Ozbiliz tied it 31 minutes later. Toby Alderweireld scored with three minutes left, but Ajax could do no more than scare United.
United is in the second-tier tournament after it was eliminated from the Champions League group stage.