Backed by a raucous crowd, Flamengo played with aggression but Bayern converted all their shots on target and reaped the rewards of a fast start.
The German Bundesliga champions struck in the sixth minute when Erick Pulgar deflected a corner kick into his own net.
Three minutes later, England striker Kane doubled the advantage with a low shot that went in off the post.
Shortly after the cooling break, the Brazilian side pulled one back as Gerson fired a missile that went in off the underside of the bar, triggering a deafening roar from the crowd.
Bayern, however, restored their two-goal lead before the interval when Leon Goretzka whipped a shot from outside the box past keeper Agustin Rossi.
Filipe Luis's side pulled one back again 10 minutes into the second half, Jorginho sending Manuel Neuer the wrong way after they were awarded a penalty for a Michael Olise handball.
The momentum had swung but Bayern held firm before finding the back of the net once more through Kane, who beat Rossi with another clinical low shot in the 73rd minute.
"It was tough. We were playing against a really good side in difficult conditions, it was really hot," Kane said.
"I thought we played well for large spells but every time we got away they clawed their way back and made it difficult. The fourth goal was important to ease the nerves and from there we managed it well."
Earlier on Sunday in Atlanta, European Champions League winners PSG sent a brutal reminder of their firepower when they cruised past Inter Miami 4-0, taking their foot off the pedal after the break in what could have turned into a humiliation for Lionel Messi's MLS team.
Joao Neves netted twice before Tomas Aviles turned Desire Doue's cross into his own net and Achraf Hakimi added the fourth, all prior to halftime.
"It was an almost perfect match, we created a lot of chances, and I'm happy yet I think we still need to improve, that's football," PSG coach Luis Enrique said.
