United moved to the brink of qualification for the knockout stages after Marcus Rashford's free-kick winner and goalkeeper Mile Svilar's blunder.
The 18-year-old debutant, the competition's youngest-ever keeper, carried Rashford's set-piece over the line having previously impressed throughout with a string of superb saves.
The win saw the Reds lead the way at the top of Group A, with three wins out of three, whilst Rui Vitoria's side remain without a point in the standings.
A draw or a victory in the return leg at Old Trafford in a fortnight's time would seal the Reds safe passage to the knockout stage.
Benfica carved out the game's first opening in the 15th minute when Alex Grimaldo picked out Eduardo Salvio, who fired narrowly wide.
Rashford headed over the bar and Svilar saved well from Nemanja Matic, before Henrikh Mkhitaryan pulled a shot wide in a flurry of chances for the visitors.
United were much improved from Saturday's dire draw at Anfield, and took the lead shortly after the hour mark in bizarre fashion.
Rashford had gone close twice with two earlier efforts from corners, and it was from another superb set-piece from the teenager that the Reds made the breakthrough.
Svilar held the inswinging 35 free-kick, but he was unable to deal with the power and accuracy and inadvertently dropped over the line with the ball.
Having alerted the VAR goal-line technology system, German referee Felix Zwayer awarded the goal.
That proved to be Rashford's last contribution as he limped off ten minutes later with a knee injury to add to United's injury worries.
Anthony Martial came on in his place and the substitute almost doubled United's lead when he burst into the box and saw his angled strike well saved by Svilar.
Ruben Dias had Benfica's one clear cut opening on 77 minutes, when he fired over from a Silvio corner when unmarked.
The hosts ended the game with ten men when veteran centre-back Luisao was sent off late on for a second booking, ruling him out of the return fixture.
The result condemned Benfica to a third straight Champions League loss, whilst United sit top of the table with a 100% record.
Basel won 2-0 in Moscow against CSKA in the other game to move up to second in the group.
Overall team performance: 7.5/10
United Faithful Man of the Match: Nemanja Matic