Bayern looked set to claim victory after Harry Kane converted a second-half penalty, but Rice found the bottom corner to ensure the points were shared. The result also extended Arsenal's impressive away record in the Champions League, with just one defeat in their last 14 matches on the road.
Arsenal suffered an early setback when Bukayo Saka was forced off injured after only 10 minutes. However, Eberechi Eze made an immediate impact after coming on, helping the visitors adjust their attacking shape.
The England international opened the scoring in the 25th minute, finishing a well-weighted Martin Odegaard pass with a half-volley into the bottom corner. Eze continued his impressive European form, having also scored in Arsenal's previous Champions League match.
Bayern responded before half-time through Alphonso Davies, who finished after Kane split the Arsenal defence with an excellent pass. The former Tottenham striker's ability to drop deep and create opportunities proved a major threat to the visitors.
Both goalkeepers produced important saves after the break, with Manuel Neuer denying Bruno Guimaraes and Eze, while David Raya kept out a dangerous Dayot Upamecano header.
Bayern eventually moved ahead when Kane converted from the penalty spot after Jurrien Timber handled the ball. The striker sent Raya the wrong way to score his 16th career goal against Arsenal.
The hosts had opportunities to extend their advantage, but Raya denied substitute Luis Diaz with a fine save. Arsenal capitalised, with Viktor Gyokeres holding off his defender before laying the ball into Rice's path. The midfielder finished brilliantly into the bottom corner to secure a valuable point.
Kane believed the result was a fair reflection of an entertaining contest, although he felt Bayern had not done enough to win.
"I thought it was a good performance, and a draw was probably a fair result overall. We just didn't quite get enough right tonight to secure the win," Kane told UEFA.
"It was nice to help get us back into the game, and I thought we had some good moments, just not enough of them over the full 90 minutes."
The England captain also discussed the upcoming Ballon d'Or ceremony, admitting that winning the prestigious individual award would be a dream come true.
"It would be amazing to win. The voting closed a long time ago, so there's nothing I can do now except wait and see what happens," he said.
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