Ashley Young and Javier Hernandez 0 win Thursday.
With a combined seven European titles, United and Ajax highlighted the last 32 pairings, and Young and Hernandez made sure the club's Champions League slump did not carry into the Europa League.
United was eliminated from the Champions League following a loss to Basel, but the road win has the three-time European champions on target to capture their first Europa League, formerly the UEFA Cup.
Four-time European champion Ajax will have to overturn the two-goal deficit at Old Trafford next Thursday to advance.
Atletico Madrid also got off to a good start on the road as Falcao scored two goals in a 3-1 win over Lazio in Rome, while last season's Champions League semifinalist Schalke needed a late goal from Klaas-Jan Huntelaar for a 1-1 tie at Viktoria Plzen.
Felipe Caicedo scored the game-winner in the 70th as Loko Moscow beat Athletic Bilbao 2-1, Adam Maher scored as AZ Alkmaar edged Anderlecht 1-0, Andre Filipe Santos scored in the 88th as Sporting Lisbon tied 2-2 at Legia Warsaw, and two goals from Jonathan Cristaldo gave Metalist a 4-0 win at Red Bull Salzburg.