Manchester, England – Aras Ozbiliz and Toby Alderweireld both 1, but couldn't overturn a 2-0 first-leg deficit in the Europa League.
Javier Hernandez scored inside 10 minutes as United built on its two goals in the Netherlands, but Ozbiliz scored late in the opening half and Alderweireld put United on its heels at Old Trafford in the 87th.
Ajax would have sealed the series on away goals if it scored again, but United was able to hold on through 2 minutes of stoppage time to advance to the final 16 of the tournament on aggregate of 3-2.
United will play Athletic Bilbao in the next round, with the first game at Old Trafford and the second leg in Spain.
Germany's Schalke squeezed into the next round as Klaas-Jan Huntelaar twice in extra time to complete a hat trick in a 3-1 win over Czech Republic's Viktoria Plzen that put his side through on aggregate of 4-2.
Huntelaar scored the opener in the 7th minute, but Frantisek Rajtoral tied the match with 2 minutes left in regulation to force the extra session. Huntelaar added goals in the 106th and 120th to seal Schalke's advancement.
Portugal's Braga followed Ajax's lead, posting a 1-0 win over Besiktas but did not advance. Turkish side Besiktas won the first leg 2-0, and squeezed through on aggregate of 2-1.
Matias Fernandez scored the lone goal as Portugal's Sporting Lisbon beat Legia Warsaw of Poland, 1-0, to advance on aggregate of 3-2. Maarten Martens scored to secure Dutch side AZ Alkmaar's progression with a 1-0 win that sealed a 2-0 aggregate advantage over Belgium's Anderlecht.
Diego Godin scored as Spain's Atletico Madrid edged Italy's Lazio, 1-0, for a 4-1 aggregate win, while Rafik Djebbour scored as Greek side Olympiacos posted its second 1-0 win to down Russia's Rubin on aggregate of 2-0.
Ukraine's Metalist Kharkiv made easy work of Austria's Red Bull Salzburg again with a 4-1 win that sealed an 8-1 aggregate victory.