Kane scored his 23rd goal of the Bundesliga season as Bayern beat Augsburg 3-1 to go nine points clear of Bayer Leverkusen ahead of their rivals' game at Heidenheim tomorrow.
Kane headed in Michael Olise's wicked cross on the hour mark to give his side the lead on what looked set to be a frustrating evening for the leaders.
The hosts had gone in front after 30 minutes when a quick free-kick caught Bayern off guard, Dimitri Giannoulis taking advantage to chest the ball inside and thrash it into the roof of the net.
Jamal Musiala converted after Leroy Sane's fine run and cross just before the break, but the game changed in the 59th minute when Cedric Zesiger received a second yellow for a challenge on Kane.
The England captain scored from the resulting attack and his side were largely comfortable from there, Sane adding gloss to the scoreline when he drilled in an injury-time third.