Lyon went ahead when Bayern's Argentina international Martin Demichelis headed in Brazilian playmaker Juninho's long range freekick for a 25th minute own goal. That was the last thing Klinsmann's team needed after two successive losses in the Bundesliga but their blushes were saved in the 52nd minute.
Miroslav Klose crossed from the right for the Roberto to head in the equalizer with Lyon keeper Hugo Lloris unable to keep the ball out despite getting a hand to it.
Klinsmann, after this improved showing compared to Bayern's last two Bundesliga offerings, denied his team had a Jekyll and Hyde quality about it. "I don't believe the team has two faces, it's in the process of developing," said the former Germany boss.
His Lyon counterpart Claude Puel reflected: "This draw is logical. After the result between Fiorentina and Steaua Bucharest (0-0) our result keeps us on track (for qualification from the group)."
Looking ahead to Lyon's next game agaisnt Steaua Puel said: "It's better to win that so as not to have to count on a rival. A victory isn't essential, but we have to continue on our journey."
Lyon's Juninho reckoned a stalemate was "a good result." Asked by French television if he had any regrets the setpiece maestro added: "Yes and no because we played against a side that put up a super second half performance, while I reckon we were better in the first."
The stalemate at the Allianz Arena lifted leaders Bayern on to four points after their opening win over Steaua Bucharest, who drew with Fiorentina on Tuesday. Lyon are third on goal difference on two points.