Bayern were three points behind leaders Borussia Dortmund when Ancelotti left but ran away with the title under Heynckes, winning 19 out of 22 league matches under his leadership to take the title with five matches to spare.
Niklas Sule's own goal put Augsburg ahead in the 18th minute, but goals by Corentin Tolisso and James Rodriguez got the champions on their way towards victory.
They could have matched Bayern's fastest championship win - under Pep Guardiola in 2014 - but a slip-up against RB Leipzig meant they had to postpone the celebrations.
"I'd like to send a greeting to Italy, to Carlo Ancelotti", said 72-year-old Heynckes, called out of retirement to step in for a fourth stint at the club.
Heynckes took over from Ancelotti after the Italian was sacked at the end of September following a 3-0 Champions League defeat at Paris St Germain.
"The coach is ultimately responsible for the big picture, but first and foremost, the footballers are the protagonists", said Heynckes.
Ancelotti responded on his Twitter account congratulating his former team as well as "friend" Heynckes after their latest title win, with the club 20 points clear at the top of the Bundesliga.
The attacking midfielder has since become invaluable once more, netting seven times so far in the Bundesliga.
"I have to think about it, too".
There is also the not insignificant matter of a DFB-Pokal semi-final against Bayer Leverkusen on April 17.
"It's always nice to win the title, but to be honest: we knew it was coming, it was nearly impossible for things to go wrong", said the Dutchman.
"We have to enjoy the moment, but it's important to keep our focus on football".
"It's always good (to win a title) - we knew it was going to happen, but we still had to confirm it", said Robben, who has now won the Bundesliga title seven times since joining Bayern in 2009.
Franck Ribery, also out of contract at the end of the season, marked his 35th birthday with a late cameo appearance off the bench.
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