The England captain opened the scoring with a penalty in the 15th minute after Union's Benedict Hollerbach fouled Michael Olise near the penalty area.
Just before the half-time whistle, Kane set up Kingsley Coman, who struck the ball past Union goalkeeper Frederik Ronnow.
Kane then completed his brace in the 51st minute, racing ahead of teammate Jamal Musiala to seize a loose ball.
Despite suffering two consecutive defeats in the Champions League against Aston Villa and Barcelona, Bayern has now scored 16 goals without conceding in their last four matches against German teams.
“There’s still a long way to go, but we’ve got the right mentality and the right attitude,” Kane told Sky.
“If we keep going like this, everything will be OK — we’re in good shape.”
Union, who needed a victory to stave off relegation on the final matchday of the previous season, currently occupy the sixth position. Captain Christopher Trimmel remarked to Sky, “we can accept this — Bayern were the favorites and possess remarkable quality.”
RB Leipzig has the opportunity to reclaim a share of the top spot later on Saturday with a win against the struggling Borussia Dortmund.
Harry Kane's two goals have propelled him ahead of his closest competitor in the race for the season's top scorer, Omar Marmoush, who netted one goal and provided two assists in Eintracht Frankfurt's 7-2 thrashing of Bochum.
Hugo Ekitike also contributed with two goals, while Marmoush, Ansgar Knauff, Nathaniel Brown, Mahmoud Dahoud, and Can Uzun all found the net during the match.
Currently, Marmoush has scored 10 goals this season, trailing Kane by just one, who has 11. However, Marmoush leads the league with six assists, surpassing Kane's five.
Frankfurt has moved ahead of reigning champions Bayer Leverkusen into third place, now six points behind leaders Bayern.
Bochum managed to score in both halves through Dani de Wit and Philipp Hofmann but were thoroughly outmatched throughout the game.
With only one point from nine matches, Bochum finds themselves at the bottom of the table, especially after both promoted teams, St Pauli and Holstein Kiel, secured victories on Saturday.
Kiel achieved their first win in the top division with a 1-0 victory over Heidenheim, courtesy of a goal from Patrick Erras, putting them three points clear of Bochum.
Goals from Oladapo Afolayan and Andreas Albers led St Pauli to a 2-0 triumph at Hoffenheim, bringing them level with their opponents on eight points, just above the relegation zone.
In other matches, Wolfsburg played to a 1-1 draw at home against Augsburg.