The 19-year-old followed up his starring role in the Premier League win over Chelsea with a well-taken strike early in the second period of extra time to give United a 2-1 win on the night at Old Trafford and a place in the last four.
An entertaining match had seen Henrikh Mkhitaryan follow up his goal in the first leg by netting his fifth goal of the competition after 10 minutes, only for Anderlecht captain Sofiane Hanni to hit back before the break.
United dominated in the second half, but Jesse Lingard, Rashford, Paul Pogba and Zlatan Ibrahimovic all squandered chances to score as the tie moved into extra time.
Victory came at a cost for Jose Mourinho, with Marcos Rojo and Ibrahimovic both taken off with injuries, but Rashford's intervention ensures their hopes of continental silverware and a place in next season's Champions League are still alive.
4 – Marcus Rashford has had a hand in four goals in his last four apps for Man Utd (all comps), as many as in his previous 33. Resurgence.— OptaJoe (@OptaJoe) April 20, 2017
There was an early scare for United when Leander Dendoncker, Anderlecht's scorer in the first leg, unleashed a dipping 30-yard strike that almost caught out Sergio Romero, only just clearing the crossbar.
But the hosts moved in front after Pogba's impressive throughball had found Rashford to start a counterattack. At the second attempt, the England forward was able to square for Mkhitaryan, who took one touch and fired low past Ruben from 15 yards.
Ruben then had to make two saves in quick succession to keep Anderlecht in the tie, getting up well to save Pogba's header before diving at full stretch to keep out an excellent curling effort from Lingard.
But United suffered a blow when Rojo was forced off on a stretcher with a knee problem to add to Mourinho's defensive injury problems – Daley Blind coming on – and the visitors almost levelled when Frank Acheampong muscled his way into the box before being denied by Romero.
Anderlecht did strike back in the 32nd minute, Youri Tielemans' 20-yard strike deflecting onto the bar and Hanni pouncing to beat Romero on the rebound from eight yards as an entertaining tie reached the break all square.
United made a strong start to the second half when Rashford and Luke Shaw combined down the left to give Lingard the opportunity from inside the six-yard box, but the stretching winger somehow failed to get enough contact on the ball to prod home.
Mourinho used his second change on the hour mark, bringing on Marouane Fellaini for Lingard, while Anderlecht introduced Nicolae Stanciu and Massimo Bruno as both sides chased the crucial a third goal.
Rashford fired wide from a good opening for United after Ibrahimovic – back in the team as one of five United changes – headed into his path.
And Rashford was guilty of another squandered opportunity when he pounced on a mistake from defender Dennis Appiah and raced clear on goal, only to overrun the ball as he attempted to round Ruben.
United were upping the pressure and it took a supreme close-range save from Ruben to deny Ibrahimovic when the striker seemed destined to score, before the former Sweden captain had a penalty claim turned down after going down under pressure from Uros Spajic.
Pogba was the next player to miss from a golden chance, volleying over with an awful attempt after Mkhitaryan's corner fell into his path six yards out, while Ibrahimovic lashed a shot wide after great work from Michael Carrick.
United's injury woes took another turn for the worse at the end of regulation time, Ibrahimovic limping down the tunnel after landing awkwardly from an aerial challenge with Kara, Anthony Martial replacing him for extra time.
Pogba had a goal ruled out in the first half of extra time, Fellaini penalised for a foul on Spajic as he headed into the Belgium international's path.
But the breakthrough finally arrived for United after 107 minutes, Rashford collecting another Fellaini knockdown before impressively working space for himself and placing a left-footed strike beyond Ruben as Old Trafford erupted.
United had to survive two last-gasp scares as Acheampong and Kara threatened with headers, the hosts holding on to ensure Martial's disallowed goal at the other end – correctly ruled out for offside - did not prove costly.
Key Opta stats:
- Jose Mourinho has won his last nine European home games as manager, including all six with Man Utd this season.
- Man Utd are unbeaten in their last 26 games in all competitions at Old Trafford (W17 D9); their longest unbeaten run since October 2011 (37 games).
- Anderlecht have never won in 18 previous away games against English sides (D2 L16), conceding in every contest.
- Henrikh Mkhitaryan has scored in five of his last six Europa League games for Man Utd.
- Only Genk (25) and Roma (24) have scored more goals in the Europa League this season than Anderlecht (23).