Vinicius gave Real the lead after 24 minutes of the first leg but Bayern scored twice in four second-half minutes to wrestle back control, Leroy Sane with a stunning opener and Harry Kane converting a penalty.
Kim Min-jae fouled Rodrygo in the box with eight minutes remaining and Vinicius stepped up, drowning out a chorus of boos and whistles to blast home.
“In this competition, it’s important not to lose, and we’re here because we haven’t lost yet,” said Vinicius.
“I’m very happy that I was able to score two goals, and now we need to have a magical night at home,” he added.
The draw makes Madrid favorites to progress in next week’s return leg at the Santiago Bernabeu but Bayern, who dominated large parts of the match, will still have hope of making it to the final at Wembley in June.
“It’s a good result for the second leg,” Madrid coach Carlo Ancelotti said. “But nothing has been decided yet.
“Bayern have high quality. They have players like Musiala or Sane who can hurt us.”
“It feels a bit strange,” said Bayern manager Thomas Tuchel.
“But the situation is very clear. Win in Madrid, then off to Wembley. The winner takes it all.
“We will be ready and we accept the fight. We will go to Madrid with self-confidence. We have to be brave.”
Before the match, the Bayern fans unveiled a giant banner depicting Franz Beckenbauer which spanned from the grass to the rafters. It was a fitting tribute on a night featuring the most-played fixture in European Cup history.
Famed for their composure on the biggest of stages in this competition, 14-time winners Real struggled early on as Bayern dominated possession while spurred on by a ferocious home crowd.
quarter-final win over Manchester City
Madrid’s success in this competition this season, including in their quarter-final win over Manchester City, has been built on absorbing pressure before striking.
The visitors repeated the trick for the opener, breaking Bayern’s dominance in a ruthlessly simple fashion while showing the hosts how easy scoring goals can be.
Toni Kroos collected the ball from a corner and drilled a defence-splitting pass along the ground which found Vinicius galloping in acres of space on the edge of the area.
Without a Bayern player in range, Vinicius calmly slotted the opener past a helpless Manuel Neuer, changing the complexion of the match completely.
Bayern, who have six European Cups of their own, were not awed and continued to push but could not break through; their best chance of the remainder of the half came through a free kick, with Kane finding a huge gap in the wall but blasting wide.
With Real seemingly in cruise control, Bayern grabbed hold of the match early in the second half, scoring twice in four minutes.
Eric Dier found Sane down the right flank. The Germany winger, who had missed several chances in the first half, dribbled into the box before unleashing an unstoppable shot. It was his first goal for Bayern in any competition since October.
The hosts’ next attack came down the left, with the ever-dangerous Jamal Musiala felled by Lucas Vazquez in the box. Referee Clement Turpin pointed straight to the spot and Kane duly sent Andriy Lunin the wrong way to take the lead.
With Bayern in control, Real went on the counter-attack and it was Kim’s turn to give away a clumsy penalty, with Vinicius confidently slotting in the equalizer.
“We have to continue with cool heads, rest until next week, and we will give everything to leave the Bernabeu qualified for London,” Vinicius said.