Bayern, who lost in the German Cup to Leverkusen in midweek, were again without injured forward Harry Kane but went in front when France defender Dayot Upamecano headed in a first-half corner in Saturday's clash.
Upamecano went from hero to villain as a woeful backpass allowed Mathias Honsak to level, only for Musiala to fire Bayern back ahead just five minutes after coming on.
Leon Goretzka and Niklas Dorsch then exchanged goals before Musiala's second sealed the win.
"I had a lot of games in a row so it was good to come on with my energy recharged," Musiala told Sky.
"Finally, I've scored at home again, it feels good. It's always nice to celebrate in front of our fans."
Bayern moved six points clear of Eintracht Frankfurt after the Eagles could only draw 2-2 at home to Augsburg.
Bayern boss Vincent Kompany started Thomas Muller in place of Kane after using Musiala as a false nine against Leverkusen.
Manuel Neuer's red card overshadowed the Cup loss and he missed the Heidenheim game with a rib injury, while Musiala was initially rested on the bench, with keeper Daniel Peretz among those coming in.
Upamecano nodded in Joshua Kimmich's corner on 18 minutes having grappled with Patrick Mainka, who was then blocked by Muller. The goal was given and the video assistant did not intervene.
Alphonso Davies struck the post for a dominant Bayern, who were then stunned early in the second half when Upamecano failed to look over his shoulder and a weak backpass was converted by Mathias Honsak.
But Musiala, who replaced Muller on 51 minutes, quickly produced the sort of magic that's made the 21-year-old one of football's top talents as he took his tally for the season to eight in stoppage time with a special goal.
In Frankfurt, another substitute Can Uzun spared the home side's blushes midway through the second half having also just entered the fray, scoring in the 74th minute to earn them a point just three minutes after Samuel Essende had put Augsburg 2-1 up.
Champions Leverkusen are a point further back in third after beating St Pauli 2-1 at home despite being without a recognised striker due to injuries.
Playmaker Florian Wirtz scored early after a clever nutmeg and finish while defender Jonathan Tah headed in a corner unmarked. Morgan Guilavogui made for a nervy finish with a goal for the Hamburg side.
"Today wasn't spectacular but we were efficient," Leverkusen coach Xabi Alonso said.
Fourth-placed RB Leipzig, who failed to win in November, followed a Cup success over Frankfurt with a 2-0 win at second-bottom Holstein Kiel thanks to Benjamin Sesko and Andre Silva's debatable penalty.
Borussia Dortmund's English winger Jamie Gittens, whore scored in last weekend's 1-1 draw with Bayern, netted another superb strike in a 1-1 draw at Borussia Monchengladbach which kept his side fifth.
Elsewhere in the German top-flight, winless bottom side Bochum lost 1-0 at home to Werder Bremen.
