Bundesliga champions Bayern Munich eased past Borussia Monchengladbach after recovering from falling behind. The thrashing left Frankfurt sixth, five points from fourth-placed Leverkusen - and a Champions League place - with four games left.
Kovac will also bring his 44-year-old brother Robert Kovac from Frankfurt as assistant coach.
Salihamidzic said: "We agreed a three-year contract on Thursday". After an 11th place finish last term, Frankfurt are all but assured of a European spot this; and it is this period of his coaching career that is perhaps most instructive as to how Kovac's Bayern might line up.
Kicker magazine reported that Bayern would pay a fee of 2.2 million euros ($2.7 million) to trigger a release clause in Kovac's existing contract with Frankfurt.
Both clubs last week denied Kovac was on his way to Munich.
Several high-profile coaches, including Tottenham boss Mauricio Pochettino, were linked with the Bayern job after Heynckes reportedly rejected the offer of a contract extension.
However, the story had already made the headlines the evening before in Germany, and Eintracht sporting director Fredi Bobic accused Bayern of leaking the news, a move he called "annoying, unprofessional and lacking respect".
Kovac is described as a team player, addressing the need to get along with the club's leaders.
"We were obviously surprised especially in this phase (of the season) we are in which is extremely important for us", Bobic said. "Not of Eintracht Frankfurt".
"Secondly, we thought for a long time about when we should tell the Frankfurters".
Bayern had appointed Kovac as he is familiar with the Bavarian side, having played for them before.
Frankfurt now lie in 6th place in the Bundesliga, five points away from Champions League qualification, with four games to go.
When the Croatian took over in March 2016, the Eagles were fighting relegation and, despite a slow start, Kovac was able to steer Frankfurt to safety - via a relegation play-off victory. Bayern's loss is Hoffenheim's gain, however, as the young coach can continue to build his team to climb the Bundesliga table. However, the performance at the BayArena did not back up Kovac's words as Frankfurt's Champions League hopes suffered a heavy blow.
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