Arsenal dominated the first half and deservedly lead at half time after Kai Havertz finished brilliantly in the 38th minute, having been played in expertly by Bukayo Saka.
Four minutes into the second half, however, Arsenal's task was made all the harder when Rice was sent off after picking up a second yellow card. Brazilian striker Joao Pedro fired Brighton level 12 minutes later.
Both sides missed golden chances to win the contest late on, but they each had to settle for a point that keeps Arsenal and Brighton unbeaten from their three games so far.
"We started the game really well," Arsenal boss Mikel Arteta told TNT Sports. "We scored the goal, 1-0. We started the second half really good but then there is the decision that changes the game completely.
"If it (second yellow card) happens throughout the game in a consistent way it is fine but it didn't. It is the inconsistency."
Rice's second yellow for trying to stop Brighton taking a quick free kick, a decision that was met with consternation around the stands, turned the tide in the visitors' favour.
Since Boxing Day 2019 - Mikel Arteta's first game in charge of Arsenal - the Gunners have been shown 16 red cards in the Premier League, at least three more than any other side. But it was Rice's first red in his 245th Premier League appearance.
Following his last-gasp winner against Manchester United last weekend, Pedro reacted first after Yankuba Minteh's shot had been saved to pull Brighton level, and the game opened up thereafter.
Saka could have snatched the win late on, while Arsenal goalkeeper David Raya had to be at his best on several occasions to keep Brighton from earning all three points, but both seemed content with a draw after an exhilarating tussle.
"We didn't create enough chances when they went down to 10 men, but we will go away and look at it," Brighton captain Lewis Dunk said.
"We always believe we can win any game of football. We are in a good place and we know we can beat anybody."
--Reuters--