The England captain equalled Rooney's total of 30 goals in the competition - the most by an English player - when he converted a 19th-minute spot-kick, then went past him with a 57th-minute finish before extending his record on Tuesday (AEST).
In the meantime, Raphael Guerreiro and Michael Olise looked to have put the hosts in complete charge by halftime at the Allianz Arena, but Bruno Petkovic and Takuya Ogiwara dragged Dinamo right back into it at 3-2.
However, Kane and Olise each struck for a second time within three minutes of each other and then the former Tottenham frontman completed his quadruple with two more penalties, before substitutes Leroy Sane and Leon Goretzka completed the rout.
Reigning champions Real Madrid needed late goals from Antonio Rudiger and substitute Endrick to finally break Stuttgart's stubborn resistance.
After a goalless first half, Kylian Mbappe had put the home side ahead, but Deniz Undav's equaliser kept the Germans in it until seven minutes from time when Rudiger and then Endrick intervened to seal a 3-1 win.
Central defenders Ibrahima Konate and Virgil van Dijk both scored as Liverpool came from behind to win 3-1 at AC Milan.
The Reds got off to a bad start when Christian Pulisic fired Milan into a third-minute lead, but a fine header from Konate dragged them back into it, before Van Dijk repeated the dose to give them an advantage at halftime.
Dominik Szoboszlai made the most of Cody Gakpo's run and cross after the break to cement the win in style.
Aston Villa paid the perfect tribute to former striker Gary Shaw as they enjoyed a dream start to their campaign with a 3-0 win at Young Boys.
Villa's return to the top table of European football was overshadowed by Monday's death of Shaw, a golden boy of the club's European Cup triumph in 1982.
Boss Unai Emery told his current crop before the game to win it for Shaw and they listened as they began life in the revamped competition in style thanks to goals from Youri Tielemans, Jacob Ramsey and Amadou Onana.
Kenan Yildiz, Weston McKennie and Nicolas Gonzalez were on target as Juventus powered their way to a 3-1 home victory over PSV Eindhoven.
Yildiz and McKennie had the hosts 2-0 up by the break, and when Gonzalez struck seven minutes into the second half, the points were safe despite substitute Ismael Saibari's stoppage-time effort for the Dutch champions.
England midfielder Angel Gomes was sent off five minutes before halftime as Lille slipped to a 2-0 defeat at Sporting Lisbon courtesy of goals from Viktor Gyokeres and Zeno Debast..Â
PA
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