Jose Mourinho's men generally looked a cut above their Belgian hosts for much of the match, but United were made to rue not taking their chances as Dendoncker salvaged a potentially vital draw four minutes from time.
Although Anderlecht initially looked bright, United soon took over and were impressive in the first half, with Sergio Romero completely untested in the visitors' goal.
The only surprising aspect of United's goal was that it took 36 minutes to arrive, with Henrikh Mkhitaryan scoring for the fourth successive Europa League away game with a straightforward finish.
Anderlecht were significantly more positive once again upon the start of the second period, but United should have finished them off when Paul Pogba spurned a late one-on-one chance.
And Dendoncker took full advantage to give the Belgians a ray of optimism ahead of the second leg at Old Trafford, where Mourinho will need more from his side amid a brutal April schedule.
1 - Leander Dendoncker scored with Anderlecht's first shot on target of the game against Manchester United. Level.— OptaJoe (@OptaJoe) April 13, 2017
Neither side looked happy to sit back and absorb pressure, resulting in an open encounter during the early exchanges, which Anderlecht slightly shaded - Frank Acheampong and Massimo Bruno shooting off-target.
But United improved quickly and the hosts were lucky to remain level in the 17th minute.
Zlatan Ibrahimovic met Marcus Rashford's teasing left-wing cross and saw his first-time effort saved by Ruben, before Jesse Lingard turned the rebound onto the post.
The lively Rashford was involved again shortly after, sending a fierce 30-yard drive agonisingly wide of the top-right corner.
United showed no sign of relinquishing their control and eventually got the away goal they deserved nine minutes before the break.
Rashford forced Ruben into a smart save from Antonio Valencia's cross, but Mkhitaryan was there to smash home the rebound.
Anderlecht were resurgent at the start of the second half and caused United problems – Valencia hooking Bruno's deflected goal-bound effort away before crucially blocking an Acheampong half-volley.
United served another warning shortly after, though, as Mhitaryan met Matteo Darmian's cross and guided an improvised effort just wide of the right-hand post.
Pogba should have wrapped things up seven minutes from the end when he smashed an effort at Ruben after a clever flicked pass courtesy of Ibrahimovic.
And the hosts made them pay.
Ivan Obradovic charged up the left flank and delivered a fine cross which Dendoncker powered onto and sent an arrowed header beyond Romero, leaving United with work to do at Old Trafford.
KEY OPTA STATS
- Anderlecht have found the back of the net in each of their last 13 European games.
- Jose Mourinho remains unbeaten against Anderlecht as a manager in Europe, drawing this game after two wins with Chelsea.
- Manchester United have scored in each of their last 10 European games.
- Only Zlatan Ibrahimovic (134) has a better mins per goal rate than Mkhitaryan (212) for Manchester United in all competitions this season (min. 2 goals).
- The Red Devils are unbeaten in their last four European away games (W2 D2), after losing each of the five before that.
- Paul Pogba attempted 127 passes in the game, the fourth-highest total by a player in a Europa League game this season (he is also second with 130 v Zorya Luhansk on September 29th).