The result was Bayern's third consecutive 1-1 draw, with the Bavarians still looking for their first win of 2023.
Union Berlin's 2-0 derby win over city rivals Hertha earlier on Saturday meant Bayern's three-point cushion on top of the table had been erased by kick-off.
"We had three or four big chances that we might have actually made the most of before Christmas," Bayern coach Julian Nagelsmann told Sky.
Bayern lacked fluency early in a testy opening half, which saw both centre-back Matthijs de Ligt and goalkeeper Yann Sommer pick up yellow cards.
With 34 minutes gone, former Man City forward Leroy Sane settled his team's nerves by hammering a cross from veteran forward Thomas Mueller past the outstretched hand of Frankfurt 'keeper Kevin Trapp.
After the goal, Bayern rekindled their early season form and produced their best spell of the match, with France defender Dayot Upamecano going close to doubling their lead just three minutes later.
Bayern continued to push for a second goal after the break but Frankfurt, who had only lost once in their previous 11 games in all competitions dating back to early October, looked dangerous on the break.
Eintracht coach Oliver Glasner brought on Japan international Daichi Kamada just past the hour and the change had an almost immediate impact, Kamada exchanging passes with Kolo Muani before the latter struck the ball past Sommer and into the net.
"We have a great group and if I can help the team, then I'm very satisfied," said Kolo Muani, who now has seven goals and 10 assists in the league this term.
"It was an intense game for us. We came here to get a point and we did it."
Union win derby
Earlier on Saturday, Union took the lead away to Hertha late in the first half through Dutch defender Danilho Doekhi, who headed in a free-kick from captain Christopher Trimmel.
Union doubled their lead halfway through the second half through midfielder Paul Seguin to run out victors over a spirited but toothless Hertha.
Visiting fans began singing "city champions" with 10 minutes remaining, celebrating their fifth consecutive derby win.
Hours after the match, Hertha announced the immediate sacking of sporting director Fredi Bobic with the club second bottom in the table.
Union coach Urs Fischer sought to play down expectations, telling AFP's sports subsidiary SID that his side "did not look like a top team".
"It was a hard-fought derby. It was close over 90 minutes. In the end, we were lucky winners. We were efficient," he said.
Elsewhere, Freiburg beat Augsburg 3-1 at home to move into fourth place.
First-half goals by towering Austrian striker Michael Gregoritsch and fellow forward Lucas Hoeler put Freiburg in the driving seat.
A 29th-minute penalty from Mergim Berisha gave the visitors hope, but Freiburg sealed their first win of 2023 through an 85th-minute strike from Philipp Lienhart.
In Bremen, Germany striker Niclas Fuellkrug scored both his side's goals in a 2-1 win over Wolfsburg, overtaking RB Leipzig forward Christopher Nkunku to top the Bundesliga scoring charts with 13 goals this season.
Promoted Bremen, who had lost six of their past eight matches came up against an in-form Wolfsburg who were unbeaten in 11 games under former Bayern coach Niko Kovac.
Borussia Moenchengladbach compounded Hoffenheim's misery with a 4-1 away win thanks to two goals and an assist from Jonas Hofmann.
Hoffenheim suffered their sixth defeat in their last eight games, leaving them just three points clear of the relegation playoff spot.
Mainz overcame the disappointment of Wednesday's last-gasp loss to Borussia Dortmund, thrashing Bochum 5-2 at home, their first win in seven league matches dating back to October.