"FC Bayern Munich have relieved head coach Niko Kovac of his duties," the club said in a statement.
The "mutual decision" was taken following talks with club president Uli Hoeness, chairman Karl-Heinz Rummenigge, sporting director Hasan Salihamidzic and Kovac himself.
"The performances of our team in recent weeks and the results have shown us that there was a need for action," said Rummenigge.
"We all regret this development. I would like to thank Niko Kovac on behalf of FC Bayern for his work, particularly for last season's double win."
Assistant coach Hansi Flick will take charge of the team "until further notice", the statement added.
The 48-year-old former Croatia international said it was the right move for the side who sit in fourth place in the table after a series of disappointing matches.
Kovac admits future in doubt after Bayern's worst loss in 10 years
"I think this is the correct decision for the club at this time," Kovac said.
"I would like to thank FC Bayern for these past 18 months. During that time our team won the Bundesliga, the DFB Cup and the Supercup. It was a good time. I wish the club and the team all the very best."
Bayern will play a Champions League match against Olympiakos on Wednesday before facing their biggest rivals Borussia Dortmund in the German top-flight on Saturday.
Saturday's hammering by Eintracht was the club's worst league defeat since a humiliating drubbing at Wolfsburg by the same scoreline in April 2009.
A 10-man Bayern was shredded by Eintracht after defender Jerome Boateng was sent off in the ninth minute, and leaving Kovac's future at the club hanging in the balance.
"I know how this business works -- I am not naive," Kovac told reporters after the match.
Bayern have won only five of their last 10 Bundesliga matches, leaving them with 18 points -- four points behind leaders Borussia Moenchengladbach.
Even their few victories have been lacklustre, with few goals to brag about and a shaky defence that has struggled to gain momentum.
"I don't think the performance we produced will bring us serious success if we don't get a handle on things," Rummenigge bluntly told the squad after a hard-won 3-2 victory at Olympiakos in late October.
Germany's top-selling Bild newspaper on Sunday reported Kovac's men had started to lose faith in him.
"The players are criticising him more often and more openly, allowing them to shift attention away from their own shortcomings," the daily wrote.
With Kovac's departure, Bayern will be hoping to turn a page on the stuttering start to the season.
Dortmund hope cup 'balm for soul' will boost Bundesliga bid
"I'm now expecting a positive development from our players and an absolute desire to perform so we can achieve our goals for the season," said Salihamidzic.
After defeating the Bavarians in the German Cup final in his last match in charge of Eintracht in 2018, Kovac went on to win the Bundesliga and the German Cup in his first year in charge at Bayern.
Only one month ago, Kovac's short-term future at Bayern looked secure after he masterminded a 7-2 win over Champions League finalists Tottenham in London.
Former Juventus boss Massimiliano Allegri, Jose Mourinho and Arsene Wenger are among the coaches named by bookmakers as possible replacements for Kovac, with the latter speaking fluent German and having previously expressed a desire to coach Bayern.
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