Marcus Rashford's extra time winner sent United through to the semi finals on a tense night at Old Trafford but victory came at a cost.
The Reds lost both Marcos Rojo and Zlatan Ibrahimovic to serious-looking knee injuries, but Rashford's brilliant 107th-minute strike finally proved decisive.
Henrikh Mkhitaryan had put United ahead but Sofiane Hanni equalised before a countless number of missed chances from the hosts led to a later-than-expected finish.
United began the night with the crucial away goals advantage and started brightly.
Leander Dendoncker flashed an effort over the bar for the Belgians, but the Reds took a nerve-settling lead in the tenth minute when Mkhitaryan continued his European hot streak.
Anderlecht keeper Ruben soon found himself called upon, and kept his side in the tie with two brilliant saves in quick succession.
First, he tipped a Paul Pogba header over the bar before a wonderful reflex save to push away Jesse Lingard's 20-yarder.
Rojo was then stretchered off to add to United's defensive injury problems, and Anderlecht capitalised with a slightly fortuitous equaliser on 32 minutes.
Youri Thielemans deflected strike came back off the bar and fell to Lukasz Teodorczyk, whose scuffed effort was turned in by Hanni.
Sergio Romero saved well from the visitors Polish striker, while at the other end Luke Shaw's ball across goal narrowly evaded the onrushing Lingard.
Substitute Marouane Fellaini went close with a header and Rashford squandered two clear openings to put the Reds ahead as United ratcheted up the pressure.
He fired wide from distance but then somehow failed to convert from six yards with the goal gaping having rounded Ruben.
Pogba lashed over and Ibrahimovic - was was particularly profligate - twice fired wide as the clock ticked down.
Rashford had a goal ruled out before Zlatan fell awkwardly on his knee and had to be taken off in the final minute of normal time.
The Reds almost went back ahead at the start of extra-time in sensational fashion.
Rashford's rabona picked out Pogba who unleashed an acrobatic overhead but Uros Spahic, who had been immense all game, blocked the effort away.
Pogba had a goal ruled out but, with a penalty shootout looming, United finally made the deserved and elusive breakthrough.
Fellaini nodded on Daley Blind's pass and Rashford pulled the ball down, created space for himself and slotted in from close range as Old Trafford erupted.
Overall team performance: 6/10
United Faithful Man of the Match: Marcus Rashford