They are now within three matches of an unprecedented unbeaten Bundesliga season on the day Bayern Munich qualified for next season's Champions League with a 2-1 victory over Eintracht Frankfurt and RB Leipzig boosted their own European hopes with a 4-1 thrashing of Borussia Dortmund. .
"I can't find the words any more, unbelievable. Our fans deserve that, they push us forward," Leverkusen captain Lukas Hradecky told broadcasters Sky on Saturday after their late escape.
Leverkusen are now unbeaten in all their 46 season matches. They still have three games left in the Bundesliga, the German Cup final against Kaiserslautern on May 25 and at least two Europa League semi-finals matches against Roma.
Stuttgart found the breakthrough early in the second half with Chris Fuhrich and increased their lead in the 57th thanks to Deniz Udav. Serhou Guirassy almost made it three, but Leverkusen pulled one back through Amine Adli in the 61st.
Odilon Kossounou thought he had the equaliser, but the goal was disallowed because Edmond Tapsoba was offside when he provided the assist.
For the second week running, Leverkusen were within seconds of losing their first game this season, but in the seventh minute of stoppage-time, Florian Wirtz sent a free-kick into the box and it bounced to the feet of Robert Andrich, who kept their impressive record alive.
"It was another wild game. Like in the other matches, we believed until the end," Andrich said.
Prolific Harry Kane scores Bayern's first in the win that secures their Champions League spot. (AP PHOTO)
Leipzig, hosting Dortmund, fell behind in the 20th minute after Jadon Sancho broke the deadlock for the guests.
But they got the equaliser just three minutes later, after a VAR review confirmed Lois Openda's goal and then took the lead when Benjamin Sesko forced a rebound from Xaver Schlager into the net shortly before the break.
Mohamed Simakan in the 46th and Christoph Baumgartner in the 80th secured Leipzig the important three points.
They're now fourth, in the last spot for the Champions League, five points ahead of fifth-placed Dortmund.
In Munich, Harry Kane slotted the ball into the net to give Bayern the lead against Frankfurt in the ninth minute, but the visitors hit back in the 23rd with Hugo Ekitike.
Kane and Bayern, however, had the final word and the England forward scored the winner from the spot in the 61st.
Bayern have qualified for the Champions League with three games remaining.
Bayern board member for sport Max Eberl said the club won't make any decision about who will replace departing coach Thomas Tuchel in summer until after the Madrid game.
Kane, meanwhile, scored his 42nd goal in his 42nd competitive game for Bayern, beating his previous personal best of 41 in a single campaign with Tottenham in 2017-18.
Elsewhere, Wolfsburg claimed a late 2-1 against 10-man Freiburg, who missed a penalty in the 87th, giving Maxence Lacroix the chance to score Wolfsburg's winner, while Werder Bremen cruised to a 3-0 win at Augsburg.
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