England's last European representatives came into the quarter-final second leg boosted by last weekend's surprisingly dominant win against Premier League leaders Chelsea and Henrikh Mkhitaryan's away goal.
Mkhitaryan struck again early on but United were given a scare, with Anderlecht captain Sofiane Hanni bringing the tie level, before Rashford capped a man-of-the-match display with the goal that secured a 2-1 extra-time triumph.
Victory saw Mourinho's men stumble a step closer to the Stockholm finale, but the Old Trafford quarter-final second leg proved costly as Marcos Rojo and Zlatan Ibrahimovic sustained what appear to be serious injuries.
Few could have foreseen such an arduous evening after Mkhitaryan's fifth goal in six Europa League matches, but impressive Anderlecht went into half-time level after Hanni reacted quickest to Youri Tielemans's deflected effort rebounding off the crossbar.
United's desperation to avoid an embarrassing exit was matched by the visitors' desire to win on English soil for the first time, leading to a thrilling second half that came to an excruciating end for Ibrahimovic.
The veteran's enforced exit after landing awkwardly saw Rashford lead the line and the outrageously-talented 19-year-old struck the decisive blow, showing wonderful composure and skill to strike a 107th-minute winner.
Few had expected such a scare from an Anderlecht side who made their ambitions clear inside three minutes, with last week's goalscorer Leander Dendoncker coming close from 30 yards.
United started as sloppily as they ended the first leg in Brussels, but showed a cutting edge so often lacking this term.
Paul Pogba's fine pass put Rashford through and the teenager fed Mkhitaryan, with United's Europa League specialist scoring with a fierce drive from the edge of the box.
Anderlecht goalkeeper Ruben had to be at his best to deny Pogba's powerful header and a Jesse Lingard strike before concern became the overriding emotion at Old Trafford.
Rojo played on after being hurt in a challenge with Alexandru Chipciu, only to go down again as Frank Acheampong raced through with Eric Bailly in tow.
Relief at seeing Acheampong turn into the side-netting was tempered by the Argentina international leaving on a stretcher, with Dendoncker and Tielemans trying to punish a rejigged backline.
They did just that in the 32nd minute.
Luke Shaw was caught deep before a cross that substitute Daley Blind cleared poorly, allowing Tielemans to get away a strike that took a deflection and rattled the woodwork. Hanni reacted quickest, slotting home a goal he could have added to before half-time had it not been for a heavy touch.
The break came as welcome relief for the hosts - but they were close to going ahead soon after the restart as Shaw surged into the box and sent in a driven cross which just evaded Lingard.
Misplaced passing and under-performing players continued to hamper United.
Mourinho turned to Marouane Fellaini, whose tame header was saved, before Rashford struck wide and raced through but took the ball too wide of goalkeeper Ruben.
Ibrahimovic was denied from close range, then saw penalty appeals waved away before he fired wide.
Rashford hit the side-netting as regulation time came to an agonising end for Ibrahimovic, whose right knee buckled when landing awkwardly.
Anthony Martial came on and extra-time was just two minutes old when Rashford's rabona set-up Pogba for an equally impressive overhead kick that was blocked.
Ivan Obradovic came close and Pogba had a goal ruled out in a thrilling first period, which came to an end with Sergio Romero jumping on a Tielemans drive at the second attempt.
Soon after play resumed, United landed the crucial blow as Fellaini nodded back for Rashford, who turned superbly and unleashed a low left-footed shot in front of the Stretford End - Ruben got a hand to the effort but could not keep it out.
However, Mourinho's men were still made to sweat.
Romero saved from close range after Acheampong ghosted in and Kara Mbodji headed over either side of a disallowed Martial goal.
The United manager's relief was clear at the final whistle - and he was quick to applaud the visiting support.