United were again made to rue missed chances as a late Leander Dendoncker header cancelled out Henrikh Mkhitaryan's potentially pivotal away goal.
Dendoncker capitalised on a defensive lapse to get in front of Matteo Darmian and power home a header four minutes from time.
Until then, the visitors had dominated almost entirely and deservedly led when Mkhitaryan slotted in on the angle after Marcus Rashford's shot was spilled.
The Armenian fired wide after the interval and Paul Pogba - otherwise hugely impressive - was denied by goalkeeper Ruben when clean through.
The hosts started brightly and went close inside the first ten minutes when Massimo Bruno - who had earlier fired just wide - had a shot blocked by the lunging Eric Bailly.
United came within inches of the breakthrough on 18 minutes.
Man of the match Rashford nonchalantly pulled down a pass and fizzed the ball in towards Zlatan.
The Swede's stabbed shot ricocheted off Ruben and fell for Jesse Lingard, who in turn could only fire against the post.
Rashford proved a thorn in the Anderlecht side throughout and went close himself moments later when his 20 yarder flashed narrowly wide.
Mkhitaryan had an effort blocked and Lingard was crowded out, but the team got the lead their attacking play deserved on 36 minutes.
A neat move that involved Ibrahimovic saw the ball moved wide to Antonio Valencia, who picked out the onrushing Rashford.
His strike was pushed away, but Mkhitaryan raced on to the loose ball to fire into the roof of the net from a tight angle - his fourth European away goal in as many games.
Anderlecht started the second half strongly and went close twice in quick succession.
Nicolae Stanciu fired narrowly over the bar from a free-kick, and Bruno's snapshot was hacked away by Valencia and the rebound was cleared.
Having weathered the mini-storm, the Reds went in search of the potentially tie-sealing second goal.
Substitute Anthony Martial was denied by Ruben and Pogba then found himself through on goal, only to be thwarted by a smart save from the keeper.
Sofiane Hanni twice went close with a deflected effort and a shot from distance, but, not for the first time, the Reds were again made to pay for their missed chances.
Dendoncker's bullet header pegged United back to leave the tie in the balance ahead of next week's return leg.
The Reds remain favourites to progress to the semi finals thanks to the away goal, but the tie is beautifully posed at the halfway stage.
Overall team performance: 7.5/10
United Faithful Man of the Match: Marcus Rashford