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Julian Nagelsmann's side drew with RB Leipzig and Cologne in its first two games back after the mid-season break but Leroy Sane's 34th-minute strike had it on course for victory on Sunday (AEDT).
That was until Kolo Muani fired home in the 69th to deny the reigning champion, which has had its lead at the top now cut to just one point.
Union Berlin's win over city rival Hertha Berlin earlier in the day put it hot on Bayern's heels, while Leipzig and Freiburg are two and three points back respectively.
Kevin Trapp produced a double save to deny Joshua Kimmich and then Thomas Muller before the latter pulled the ball back for Sane three minutes later to sweep home an opener.
Bayern defender Dayot Upamecano headed wide from close range and Trapp kept out Sane's next effort as the host pressed for a second.
But unlike on the opening day of the season, when it was five goals up by half-time in a 6-1 win over Eintracht, Bayern lacked a finishing touch.
That proved costly as, from Eintracht's first on-target attempt, Muani fired across Yann Sommer after powering past Upamecano.
Bayern brought on Serge Gnabry – among the substitutes after being criticised for attending Paris Fashion Week – but Eintracht looked the more likely to find a late winner late on as Rafael Santos Borre's deflected effort flew just wide after Evan Ndicka's close-range attempt was blocked.
