Gladbach also hit the post in an astonishing match against opponents who played for more than one hour with 10 men after Cleber was sent off.
Anthony Modeste, the league's leading marksman, scored two more goals to give Cologne a 2-1 win over basement side Ingolstadt, taking his tally to seven and lifting the Billy Goats to second place, two points behind Bayern.
Bayern coach Carlo Ancelotti was angry with his team's display against Eintracht and promised to make changes.
"We didn't play well and showed the wrong attitude," said the Italian.
"We went to sleep for the first 45 minutes and that is too much in any game. We can lose concentration for a few minutes, that can happen, but not for that long. I will change that, I will make changes to the team."
Arjen Robben gave Bayern an 11th-minute lead but the hosts levelled two minutes before halftime when Szabolcs Huszti fired past Manuel Neuer for his team's first goal against the Bavarians since 2011.
Joshua Kimmich put Bayern back in front from a corner in the 62nd minute and the hosts were in more trouble when Huszti was sent off for a second yellow card.
But Frankfurt levelled in the 78th minute when Marco Fabian turned in Timothy Chandler's cross shot with his chest as frustrated Bayern, who won their first five games, drew for the second match in a row.
Gladbach seemed to be in the driving seat when Hamburg's Brazilian defender Cleber gave away a penalty and was sent off in the 25th minute, but Andre Hahn's spot kick was brilliantly saved by Rene Adler.
The Foals won a second penalty on the hour after Douglas brought down Ibrahima Traore and this time Lars Stindl took the kick but sent his effort against the crossbar.
Adler denied Fabian Johnson and Oscar Wendt hit the post for the Foals as Hamburg, one place off the bottom, picked up only their second point of the season in unlikely circumstances.
Modeste scored twice in a 11-minute first-half spell, the second from a penalty, to set up Cologne's win although there was a nervous finish after Lukas Hinterseer pulled one back for Ingolstadt, also from a penalty, late in the game.
Nineteen-year-old forward Ousman Manneh scored the decisive goal to give Werder Bremen a 2-1 win over Bayer Leverkusen.
The Gambian slid in to score his first Bundesliga goal on his fourth outing for the club following a well-worked free kick as Bremen completed three games without defeat.
There was another penalty at Hoffenheim where Andrej Kramaric converted in the 81st minute to give Hoffenheim a 2-1 win over promoted Freiburg.
Augsburg held Schalke 04 1-1 in the other afternoon kickoff.
(Writing by Brian Homewood in Berne, editing by Pritha Sarkar)
