Tah was left completely unmarked to head in from close range and complete the comeback for the Bundesliga leaders after they had twice cancelled out Stuttgart goals.
Leverkusen are the first team in Europe’s top five leagues to reach 30 games unbeaten in all competitions from the start of the season since Juventus in 2011-12.
“Obviously this win gives us a big push, a lot of energy for the task ahead of us [on Saturday],” Tah said. “We are looking forward to it.”
Leverkusen, whose most recent major title victory was the 1993 DFB Pokal, are still in the mix for three titles this season, including the Europa League.
“What makes us strong is that we are focusing only on us,” Tah said. “We are not focusing on what could happen at the end of the season, and that is the we way we will keep going.”
The visitors took an 11th minute lead with Waldemar Anton‘s looping header at the far post after he shook off his marker.
The hosts bounced back five minutes after the restart with Robert Andrich curling an unstoppable shot into the top corner but in the 58th he lost possession allowing Chris Führich to drill in for a renewed Stuttgart lead.
With the game set for extra-time, Germany international Wirtz floated a deep, pin-point cross for Tah to snatch the winner.
The Associated Press and Reuters contributed to this report.