France twice led through Alexandre Lacazette, who netted either side of Timo Werner's leveller, but Stindl struck in stoppage time to rescue the hosts.
Germany, whose last defeat came against France in the semi-final of Euro 2016, have now failed to beat them at home since 1987.
Both teams have qualified for next year's World Cup in Russia and are considered among the title favourites.
Germany coach Joachim Loew was eager to test his squad and made six changes to the team that drew 0-0 against England at Wembley on Friday, including three in defence.
But the hosts struggled under early French pressure, with the speedy Kylian Mbappe and Blaise Matuidi a constant threat.
Lacazette, replacing injured forward Olivier Giroud, forced a good save from Germany keeper Kevin Trapp with a powerful shot but went one better in the 34th minute, tapping in from close range after superb work from Anthony Martial.
"In the second half we managed to play more freely," Loew said. "We were able to find the spaces, use them better. I told them at halftime to stay focused, keep working at it.
"Overall I am extremely satisfied with our two friendly matches."
The Germans woke up after the break and substitute Antonio Rudiger should have levelled when he was set up by Julian Draxler in the box.
Instead it was Werner who grabbed the equaliser in the 56th minute with his seventh goal in his 10th international as he finished off a quick break launched by Mesut Ozil.
Toni Kroos then hit the bar in the 70th as the Germans looked to be in complete control before another quick French break earned the hosts their second goal.
Martial should have added a third late in the game but Trapp did well to deny him as the forward charged into the box before Ozil played another through ball to Mario Goetze, who in turn set up Stindl for his added-time goal to salvage a draw.
France coach Didier Deschamps praised his team's speed and skill but said the Germans were still ahead of them in terms of quality.
"We were closer to scoring a goal than conceding," Deschamps told reporters. "We did a lot of good things against this German team. We had speed and skill. But they caused us problems too.
"They are a high-class team... We are close to this team but they are still ahead of us."
(Reporting by Karolos Grohmann; Editing by Ken Ferris and Toby Davis)
