The German champions said that 65-year-old Heynckes, currently leading Bayer Leverkusen's challenge for the Bundesliga, would replace Van Gaal at the end of the season on a two-year contract.
The club had already announced Van Gaal would be leaving following a disappointing campaign.
Heynckes, whose low-key player-friendly approach is the complete opposite to the explosive and colourful Van Gaal, was previously at the helm from 1987 to 1991 when he won two Bundesliga titles and again in 2009 on a caretaker basis.
"Jupp Heynckes was the candidate we were wishing for," Bayern Munich CEO Karl-Heinz Rummenigge said on the club website. "We're very fortunate.
"I'm convinced that we will can very successfully work together with Heynckes."
Heynckes said: "Naturally, when you are Bayern coach you are to a certain extent duty-bound to be successful."
Bayern have alternated between adventurous coaches, such as Jurgen Klinsmann, and more orthodox names, such as Ottmar Hitzfeld, in the past few years.
Leverkusen are second in the Bundesliga but Heynckes had already agreed to leave at the end of the season despite their success.
Last season, Leverkusen equalled the record for the longest unbeaten start to the Bundesliga season when they went 24 games without defeat, although they eventually finished fourth.
The previous record had been set by Bayern, also coached by Heynckes, in 1988-89.
Heynckes is one of Germany's most travelled and long-serving coaches, having also worked at Athletic Bilbao, Real Madrid, Tenerife, Schalke 04, Eintracht Frankfurt and Benfica.
He began his coaching career with Borussia Moenchengladbach in 1979, where he also ended his playing career, and stayed there for another eight years.
One of his biggest achievements was at Real where they won the Champions League in 1998, ending their 32-year wait to win Europe's top competition for the seventh time.
He was still sacked that season because Real did not win the Spanish league.
Van Gaal won a league and cup double last term and led Bayern to the final of the Champions League where they lost to Inter Milan, who also beat them in the last 16 of this season's competition.
His contract was extended until 2012 in September despite a stuttering start but, in a move typical of Germany's coaching merry-go-around, he agreed earlier this month to leave at the end of the campaign with his side fourth in the Bundesliga.
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