The 38-year-old rushed out of the box in typical fashion 18 minutes in to the German Cup last-16 clash, taking out an advancing Jeremie Frimpong and seeing a straight red.
Neuer's sending-off brought Israel goalkeeper Daniel Peretz on for his Bayern debut, 18 months after joining the club.
Bayern, who were already without the injured Harry Kane, dug deep despite the one-man disadvantage and dominated the ball.
The visitors eventually broke through in the 69th minute, though, with Tella heading in an Alex Grimaldo cross.
Tella had only been on the field for eight minutes having come on for injured striker Patrik Schick, who himself was subbed on at half-time.
Home winger Michael Olise lashed inches wide in the fourth minute of stoppage time but Leverkusen held on to inflict a first domestic defeat on Vincent Kompany since he became Bayern coach.
"We were too hectic and tried to go forward too quickly," Leverkusen captain Jonathan Tah told ARD. "At half-time, we said we need to be patient -- and we did it.
"I didn't know Nathan could be so good with his head."
Neuer told reporters: "The red card decided the game. It hurt us and I'm sorry."
Xabi Alonso's Leverkusen, who won an unbeaten league and cup double last season, are through to the last eight.
Bayern have won the German Cup a record 20 times -- 14 more than the next best -- but have not made it past the quarter-finals since 2020, when they defeated Leverkusen 4-2 in the final.
Elsewhere, a stoppage-time goal from Anthony Jung took Werder Bremen to the quarters with a 1-0 win over Darmstadt.
Stuttgart are also through after a 3-0 win at Regensburg, while Freiburg were knocked out by a 3-1 loss at Arminia Bielefeld.
Three-time winners Stuttgart were dominant against their second-division opponents, taking the lead just 10 minutes in thanks to a clever individual goal from Enzo Millot.
Japanese defender Anrie Chase headed in from a free-kick nine minutes later to double the visitors' lead -- his first goal for Stuttgart.
Nick Woltemade broke through and rounded the goalkeeper to score Stuttgart's third.
Beaten finalists in 2022, Freiburg were heavy favourites to beat third-division Bielefeld and looked on course when Michael Gregoritsch won a penalty 15 minutes in, drawing a clumsy foul from Maximilian Grosser.
But Florent Muslija's tame spot-kick was easily saved.
The hosts then took the lead thanks to an excellent long-range effort from Christopher Lannert on 28 minutes, before Gregoritsch gave away a penalty, which was nervelessly converted by Julian Kania nine minutes before the break.
Gregoritsch pulled one back just after the hour, but Louis Oppie put Bielefeld out of reach with a goal nine minutes from time.
On Wednesday, RB Leipzig -- winners in 2022 and 2023 -- host five-time winners Eintracht Frankfurt.
