The six-foot 185-pound Wenger, who can play defence, midfield and up front, was selected first overall at the draft in Kansas City on Thursday, adding another piece to the Impact lineup heading into their first MLS season.
Goal-scorer Darren Mattocks of the University of Akron went second overall to the Vancouver Whitecaps.
''We just felt he has the total package,'' Impact sporting director Nick de Santis said of Wenger. "Mattocks and couple of other players were there and it was tough to choose, but we felt good about the decision. In athleticism, he's a notch above everyone.''
Canadian teams were front and centre at the draft, as Toronto FC used the fourth overall pick to take UC Santa Barbara midfielder Luis Silva and then got Notre Dame defender Aaron Maund 12th overall.
Wenger won the Hermann Trophy as the top player in U.S. university soccer last season.
The 21-year-old anchored Duke's back line in his first two seasons. He was converted to forward in his third season, scoring 17 goals and adding eight assists to lead the Blue Devils to an 11-8-3 record. Wenger said he is most comfortable in the midfield but is ready to play any position for Montreal.
''It was a surprise. Going No. 1 was quite an honour,'' he said. ''It was more important for me to be picked into a fantastic organization that would help me grow as a player and give me all the opportunities and resources necessary to succeed, and I think Montreal will do a fantastic job.''
Montreal was expected to take Mattocks because they are thin up front, but coach Jesse Marsch and his staff felt they couldn't pass up Wenger.
''I think Andrew Wenger has the potential to become a great midfielder, defender and forward,'' said Marsch. ''He was the best soccer player available.
''We followed him very closely, and we just felt he had the best future, short term and long term, in the draft.''
Wenger may miss stretches of his rookie season on U.S. national teams duty and there are reports he may stay on at Duke until May to complete his semester.
''We truly believe in him finishing his school as well,'' said De Santis. ''We're not going to pressure him to come in the first minute and perform.''
He will miss the start of camp on Sunday because he is attending the U.S. under-23 team camp in California, where one of his teammates is Impact defender Zarek Valentin.
Wenger, of Lititz, Pa., has known Valentin since they were children.
''We played together from 11 until we went to college,'' Wenger said. ''He's currently my roommate at the U.S. under-23 national team camp, so we've been talking about Montreal and different scenarios over the last week or so.
''He's already been in the league for a year, so he's going to be able to show me what it is to be a professional and show me the ropes. It's a great advantage for me.''
The Whitecaps were delighted to land Mattocks, who had 39 goals in 47 matches in the last two seasons at Akron.
''He brings different qualities to the front line,'' said Whitecaps coach Martin Rennie, who already has skill up front in Camilo Sanvezzo and Eric Hassli. ''He's more of a striker who gets behind the defence. He stretches the defence. And he's got finish.
''He's a prolific goal-scorer and that's hard to find.''
It was the second straight year Vancouver took a striker in the draft. As an expansion club last year, the Whitecaps picked Omar Salgado first overall. Rennie said the two may make a strong pair up front in future.
Mattocks felt no disappointment at going second to Wenger.
''Wherever I go, whether it was first or 10th, I just want to stay humble and do my best,'' said Mattocks, who had 21 goals in 22 games last season.
Silva scored 17 goals in 22 games at Santa Barbara and is seen by TFC as a midfield playmaker.
''I'm a team player, I work hard both sides of the ball,'' said Mexican-born Silva. ''I just know (TFC) is an attacking team with the 4-4-3 formation, which I'm used to playing.
"I've played every formation. It's a matter of going to training and seeing how the coach wants to use me.''
TFC coach Aron Winter sees Silva as a potential midfield general who can also be used on the wing, while Maund brings size and quickness to the defence.
''The reason we drafted them is because they're good kids, with good technical qualities,'' said Winter.
In the second round, Montreal took midfielder Calum Mallace of Marquette 20th overall and Vancouver picked defender Chris Estridge of Indiana 21st. Toronto FC did not have a second-round selection.
The New England Revolution took UCLA midfielder Kelyn Rowe with the third pick. Chivas USA took Maryland forward Casey Townsend fifth and the San Jose Earthquakes took Silva's Santa Barbara teammate Sam Garza sixth.
A total of 38 players were selected in two rounds.