Gunners midfielder Rice lashed low beyond Bayern goalkeeper Manuel Neuer eight minutes from time to extend his side’s unbeaten start to this season’s competition following back-to-back wins over Napoli and Lille.
Kane – Tottenham’s all-time record scorer – was on course to claim the winner after registering his 16th goal in 23 career outings against Arsenal by converting a 61st-minute penalty following Jurrien Timber’s handball.
The England captain, who is contention to win the Ballon d’Or on Monday, earlier produced a sublime assist for Alphonso Davies to cancel out Eberechi Eze’s opener.
But in their biggest test since losing last season’s final to Paris St Germain on penalties, Mikel Arteta’s Premier League champions responded well to deservedly return home with a share of the spoils.
A pulsating draw in Bavaria was soured slightly by an early injury suffered by Bukayo Saka, while the result ended Arsenal’s run of 14 successive Champions League group-stage victories.
Noni Madueke was recalled in place of Eze following his late winner at Nottingham Forest on Sunday, with Timber coming in for Ben White in the Gunners’ other change.
Meanwhile, Bayern boss Vincent Kompany was fit enough to take his place on the touchline after a viral infection.
Arsenal controlled the early exchanges before suffering a blow when Saka limped off to be replaced by Eze in the 10th minute.
With Michael Olise suspended, Jamal Musiala injured and Luis Diaz only on the bench, Bayern’s attacking fear factor was significantly diminished.
Riccardo Calafiori threatened from a Martin Odegaard corner before Arsenal were almost immediately caught out on the counter-attack when a goal-kick taken by the quick-thinking Kane culminated in David Raya saving well from the unmarked Serge Gnabry.
Arsenal continued to dictate proceedings and edged ahead in the 25th minute. Eze was sent racing into Bayern’s 18-yard box by Odegaard’s delightful ball over the top before rifling a left-footed shot beyond Neuer and into the bottom-right corner.
Celebrations were briefly halted by an offside flag, only for VAR to overturn the decision, to the delight of almost 3,800 travelling fans high up at the other end of the pitch.
However, Arsenal’s lead lasted just 10 minutes.
Kane, who spent much of the game dropping deep, was the architect, producing a sumptuous through ball from inside the centre circle for Davies to outpace Timber and slot past Raya.
For all of their possession, Arsenal appeared vulnerable on transitions and could have gone into the break behind as Kane had an effort blocked before Gnabry picked the wrong option by shooting from distance.
A breathless second half began with a superb save at either end.
First Neuer turned Bruno Guimaraes’ header onto the woodwork following a Rice corner, before Raya superbly repelled after Dayot Upamecano nodded goalwards.
Kane put Bayern ahead with 29 minutes remaining, dispatching his spot-kick high into the left of the net at the end of a stuttering run-up after seeing a shot blocked by the arm of Timber.
Konrad Laimer lashed wide as Bayern came close to doubling their advantage before Rice salvaged a point for Arsenal by latching on to Viktor Gyokeres’ pass to slam into the bottom-left corner.
The Gunners almost grabbed a late winner but Timber’s effort was tipped over by Neuer as an entertaining encounter ended all square.