Thomas Mueller scored from the spot before Gomez scored twice in the second period to put Bayern three ahead.
Martin Harnick managed to pull two back for Stuttgart but Bayern held on.
The Bavarian club, who will be taken over by former Barcelona coach Pep Guardiola on 26 June, finished the season with 15 wins from their final 16 matches and have only suffered one defeat since October.
And after the match Heynckes said: "I have decided to give a closing press conference on Tuesday and then I will reveal my future plans."
Heynckes, 68, had previously said he was too old to coach overseas but he has been linked with a return to Real Madrid, who he coached to the 1998 Champions League title at the end of his single season as their coach.
"It's unbelievable what the team and coach have achieved over the whole season,'' said Bayern captain Philipp Lahm.
"They were tough months but now we're being rewarded for them.''
Only Celtic in 1967, Ajax in 1972, PSV Eindhoven in 1988, Manchester United in 1999, Guardiola's Barcelona in 2009, and Inter Milan in 2010 previously won the treble.
Mueller set Bayern on their way with a penalty after Lahm was tripped by Ibrahima Traore.
After the break Gomez converted crosses from Lahm and Mueller.
Austrian international Harnick scored with a fine header from Gotoku Sakai's cross before setting up a tense finish with a firm shot after he was denied by Bayern keeper Manuel Neuer.
But the Bundesliga and European champions withstood heavy late pressure to complete the treble and send Heynckes out on a high.
