Musiala seized on defensive errors to score his club's third-fastest Bundesliga goal ever and Bayern were 3-0 up after just 13 minutes thanks to a precise finish from Kane and an own goal from Kiel's Nicolai Remberg.
Kane added another just before halftime and Michael Olise scored on a rebound in the 65th for his first goal since joining Bayern from Crystal Palace.
Armin Gigovic salvaged some consolation with an 82nd minute goal for Kiel, who have lost all three of their league games since promotion.
England captain Kane, who scored twice against Finland on his 100th international appearance on Tuesday, then completed his perfect week by completing his hat-trick from the penalty spot in added time after he was brought down by Carl Johansson.
It was Kane's 40th Bundesliga goal from 35 games since joining the club last year.
Bayern are now on a four-game winning run since Kompany took over for this season - three victories in the Bundesliga and one in the German Cup.
Victor Boniface scored twice as champions Bayer Leverkusen beat Hoffenheim 4-1 to rebound from a rare Bundesliga loss, but Leipzig dropped points for the first time this season in a 0-0 draw with Union Berlin.
Leverkusen got back to winning ways after losing to Leipzig 3-2 in their last game - the first time Xabi Alonso's title-winning team had been beaten in the Bundesliga for 15 months.
Boniface set up Martin Terrier for Leverkusen's first goal, and scored himself soon after, leaving defender Alexander Prass sprawled on the ground as he scored past goalkeeper Oliver Baumann.
Leverkusen's three games in the league have featured a combined 15 goals, in part because of lapses in concentration in defence which have frustrated Alonso.
Mergim Berisha's goal in the 37th minute cut Leverkusen's lead to 2-1 and the champs nearly conceded another soon after.
But they retook control after the break, a penalty from Florian Wirtz and another goal from Boniface ensuring the 4-1 win.
It was the first time Boniface had scored twice in a game since an adductor muscle injury ruled the Nigerian striker out of much of the second half of last season.
Union goalkeeper Frederik Ronnow frustrated Leipzig in a goalless draw, saving a penalty from Lois Openda and a stoppage-time free kick from David Raum.
New signing Ermedin Demirovic scored twice as Stuttgart beat Borussia Monchengladbach 3-1, while Eintracht Frankfurt beat Wolfsburg 2-1 thanks to two goals from Omar Marmoush.
Junior Adamu scored twice in three minutes to give Freiburg a comeback 2-1 win over Bochum.
