United showed great attacking desire in the early stages as Rashford, Depay and Jesse Lingard all threatened but, for all their dominance of possession, the hosts found themselves behind in the 27th minute.
Marco Urena fed Sisto, who jinked away from Daley Blind and Michael Carrick on the edge of the box before firing past Sergio Romero into the bottom-left corner.
Depay shot straight at Andersen as United went in search of an immediate response that duly arrived when the Dutchman got to the byline and drilled in a low ball that Bodurov diverted home.
Morgan Schneiderlin went close with an acrobatic effort and Depay shot narrowly wide from 25 yards before Schneiderlin headed against the post as United pressed hard to level the tie before the break.
Romer’s foul on Herrera gave Mata the perfect opportunity to do just that, but the Spaniard saw his penalty pushed behind by Andersen diving low to his right.
United began the second half brightly and Schneiderlin opted to square instead of shoot from Mata’s lofted ball, before Herrera headed a golden chance into the side netting following more trickery from Depay.
Shortly after the hour, Mata did well to keep Varela’s cross in next to the left byline and cut the ball back for Rashford, who converted with a minimum of fuss from eight yards.
And, after Lingard was booked for a dive that will keep him out of the first leg of United’s last-16 tie, Rashford was on hand to sweep home Varela’s cross from the right to complete his double.
But the drama did not end there as Herrera converted from 12 yards following Kian Hansen’s handling of Depay’s shot three minutes from time.
Depay was then hauled back by Romer, who was issued with a second caution, and the Dutchman capped a fine performance by rifling in the fifth as United romped to victory and eased some of the pressure on Van Gaal.