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 also maintained their unbeaten run across all competitions this season.
"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."
Xabi Alonso's team, whose most recent major title victory was the 1993 German Cup, 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."
Leverkusen, who face Bayern Munich in the league on Saturday, maintained their unbeaten run across all competitions this season.
Fortuna Dusseldorf and Kaiserslautern are already through to the last four while Saarbruecken face Borussia Moenchengladbach in the remaining quarter-final on Wednesday.
