Benfica had an early penalty shout waved away when Eric Bailly shouldered Pizzi off the ball, but United were handed a penalty at the other end scarcely a minute later when Douglas' hand fell on the ball.
However, Anthony Martial failed from the spot as his soft shot was saved by 18-year-old goalkeeper Svilar, whose error in the previous matchday allowed United to win in Portugal.
Soon after, United keeper David De Gea was called on to make a fine save of his own as he flew to his right to paw aside a hard shot from Diogo Goncalves.
But United found the breakthrough before the half when Nemanja Matic unleashed a shot from 30 yards that came back soundly off the post, only to strike the back of Svilar -- who have dived for it late -- and fall over the goal line.
Svilar was unfazed, though, and raced out to deny Romelu Lukaku on the break in first-half stoppage time.
Jesse Lingard picked up a knock and was replaced by Henrikh Mkhitaryan at half-time.
Only another fine, diving save from De Gea preserved United's lead on the hour mark as the keeper got his fingertips to Goncalves' low shot, and five minutes later United were fortunate again as Bailly gave the ball away at the back, only for Raul Jimenez to strike the post.
Marcus Rashford came on as a substitute on his 20th birthday and within three minutes drew a penalty from an Andreas Samaris foul. Daley Blind stepped up to the spot and beat Svilar to double United's lead.