Bayern dominated but Madrid survived, and the tie is still wide open ahead of the second leg in Madrid on May 8.
With the focus on England stars Harry Kane and Jude Bellingham, as well as Toni Kroos's return to his former team, Vinicius emerged as the decisive figure on a night when his bright green boots were often a blur.
The Brazil forward opened the scoring on Tuesday (local time) against the run of play in the 24th minute and equalised with a coolly taken penalty in the 83rd after Leroy Sane and Kane had scored for Bayern.
Kane's goal, another penalty, was the England forward's 43rd goal in 43 games for Bayern this season.
The winners of the two-leg semi-final will play either Borussia Dortmund or Paris Saint-Germain in the final in London on June 1.
Bayern totally dominated the opening stages. Sane had an effort saved by Andriy Lunin in the first minute, and the hosts went on to miss a slew of chances.
Nearly 20 minutes elapsed before Madrid enjoyed a period with the ball.
Then Kroos spotted VinÃcius's run and played a perfect through ball for the Brazilian to sweep the visitors' first real chance past Manuel Neuer in the Bayern goal.
Bayern lost their early zest as the visitors saw more of the ball. Bayern coach Thomas Tuchel gesticulated wildly at his players to push forward, then threw his arms up in exasperation.
Kane went closest with a free kick before the break, awarded for a foul by Nacho on Jamal Musiala, who gingerly rubbed his bandaged left leg. The Germany forward had only just recovered from a knee problem.
Sane equalised in the 53rd minute when he ran at the Madrid defence and cut inside Ferland Mendy before firing a fierce strike just inside the near post.
Kane made it 2-1 four minutes later with a penalty awarded after Lucas Vazquez bundled over Musiala.
But it wasn't the last penalty as Kim Min-jae brought down Rodrygo, and Vinicius duly equalised from the spot.
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