Augsburg was beaten 4-1 away for Bayern to seal its record 28th German championship, including the solitary pre-Bundesliga title in 1932.
Heynckes could yet cap a fairytale return by winning the European Cup for the sixth time in Bayern's history in the Champions League final in Kiev on May 26.
"It's a special moment, because we now know we're champions", said Ribery, who has now won the Bundesliga eight times since joining in 2007.
Augsburg took a shock lead through an own goal by Niklas Suele, but the Bavarians roared back as Corentin Tolisso, James - who then assisted Arjen Robben - and Sandro Wagner all scored.
Heynckes also claimed that he "never imagined" Bayern would win the league this season when he was called upon to take over at the Allianz Arena back in October, insisting that his priority was just to steady the ship in Bavaria. "Everyone delivered great work".
Kimmich remained in the thick of things and set up another goal after dancing through Augsburg's defence before finding Juan Bernat, whose back-heel pass allowed James Rodriguez to tap home the 2-1 at the far post six minutes later. "This title is also for him", he added. "He is a great coach", Bayern's head coach Jupp Heynckes said.
Cordova's shot was blocked by Bayern goalkeeper Sven Ulreich, but the ball ricocheted up into Suele's face and into Bayern's goal to stun the visitors on 18 minutes.
Wagner, playing in place of the rested Robert Lewandowski, rounded off the scoring late.
Hamburg suffered an early setback when Naldo appeared to score with his hand, but goals from Filip Kostic and Lewis Holby put the home side ahead before Guido Burgstaller's equalizer for Schalke. The win moved the Wolves two points clear of Mainz 05 who stayed in the relegation playoff spot after drawing 1-1 at Cologne.
An early goal by Cologne's Germany defender Jonas Hector was cancelled out as Pablo De Blasis headed an equaliser for Mainz five minutes into the second half.