The Europa League has acquired huge significance for United and manager Jose Mourinho as it offers a potential route into the Champions League, away from the battle for top-four places in the Premier League - making this victory crucial.
United took the lead on the night and in the tie when Henrikh Mkhitaryan drilled in a low finish in the 10th minute but Anderlecht restored parity when Sofiane Hanni scrambled home an equaliser after 32 minutes.
Mourinho's side were their own worst enemies with a shocking display of finishing as they missed chance after chance, their cause also undermined by injuries to defender Marcos Rojo in the first half and a serious-looking knee injury to top scorer Zlatan Ibrahimovic at the end of normal time.
United were facing the prospect of a penalty shootout but Rashford, a scorer against Chelsea at the weekend, made the decisive contribution after 107 minutes with a brilliant turn and finish from Marouane Fellaini's knockdown.
Manchester United flirted with an exit from the Europa League here - and if they had gone out they would only have had themselves to blame.
As on so many occasions this season, United created multiple chances only to waste the opportunities and leave themselves hostages to fortune and the threatening counter-attacks of Anderlecht.
Rashford, Ibrahimovic and Paul Pogba were all guilty of a succession of bad misses as the flaw that has undermined United all season reared its ugly head once more and kept Anderlecht in contention right until the final whistle in extra time.
On this occasion, at least, United rescued themselves with Rashford's goal but Mourinho will know his side must discover the killer touch from somewhere if they are to secure the Champions League place that must be the minimum requirement from this season.
Manchester United switch their focus back to the Premier League as they travel to Burnley on Sunday (14:15 BST) before travelling to Manchester City on 27 April (20:00).