Spurs led 2-0 as Erik Lamela slid in to convert Georges-Kevin Nkoudou’s low cross at the back post and Christian Eriksen drilled in a second for Spurs.
But Bayern came back thanks to goals from teenagers Jann-Fiete Arp and Alphonso Davies, who levelled with a fantastic 25-yard curling effort.
Paulo Gazzaniga saved from David Alaba and Jerome Boateng in the shootout.
Spurs lost defender Juan Foyth in the second half to a serious-looking ankle injury and he left on a stretcher.
Spurs looked in control at 2-0 and Son Heung-min went close twice, forcing two good saves from Sven Ulreich, with the goalkeeper also denying Harry Kane, who had set up Eriksen’s second.
Spurs won the shootout 6-5 with Eriksen the only Tottenham player to miss his kick.
Benzema scores hat-trick in Real Madrid win
Real Madrid finally picked up their first win of pre-season, at the fifth attempt, as they beat Fenerbahce 5-3 in the third-place play-off thanks to Karim Benzema’s hat-trick.
Garry Rodrigues gave Fener, who lost 6-1 to Bayern Munich on Tuesday, an early lead from the edge of the box.
Benzema converted Vinicius Jr’s cross to level and then headed in from Toni Kroos’ free-kick.
Nabil Dirar’s deflected shot got the Turkish side level – but Benzema then headed in Lucas Vazquez’s cross for his third.
Ozan Tufan scored for Fenerbahce but Nacho and Mariano gave Real Madrid the win.