Bayern took the lead through Lennart Karl before former Bayern midfielder Maurice Krattenmacher levelled for Elversberg. Kane then restored the champions' advantage in the 72nd minute to secure three important points.
• Elversberg: Maurice Krattenmacher 61'
• Bayern Munich: Lennart Karl 26', Harry Kane 72'
Elversberg began confidently despite facing the reigning Bundesliga champions and created several promising moments in the opening stages.
Bayern eventually found the breakthrough in the 26th minute. Ismaël Saibari's cross travelled across the face of goal and reached Karl at the far post, with the youngster making no mistake to put the visitors ahead.
Elversberg thought they had equalised shortly before half-time when Cole Campbell found the net, but the effort was ruled out for offside after a VAR review. Bayern therefore went into the interval with a narrow lead.
The home side continued to push after the restart and were rewarded in the 61st minute. Maurice Krattenmacher, who joined Elversberg from Bayern during the summer, produced a superb curling effort from outside the box to beat Jonas Urbig and make it 1-1.
Bayern responded by increasing the attacking pressure. Vincent Kompany introduced Joshua Kimmich, Jamal Musiala and Luis Díaz, and the changes helped the visitors regain control of the contest.
Michael Olise came close to restoring Bayern's lead when his effort struck the crossbar, but the decisive moment arrived two minutes later.
In the 72nd minute, Aleksandar Pavlović picked out Kane inside the penalty area, and the England captain produced a composed finish past the goalkeeper and two defenders to put Bayern 2-1 ahead.
The goal was Kane's first Bundesliga strike of the season and, according to reports, took him to 100 Bundesliga goals.
Elversberg refused to give up and continued searching for another equaliser, but Bayern's defence held firm during the closing stages.
The result leaves Bayern on seven points from three Bundesliga matches, two points behind early leaders Freiburg and Borussia Dortmund. Elversberg remain on six points after suffering their first defeat of the campaign following impressive victories over Bayer Leverkusen and Borussia Mönchengladbach.
For Bayern, the victory was more about resilience than brilliance. The champions were forced to work hard for the three points, but Kane once again demonstrated his importance by delivering when his team needed him most.