The Gunners appeared to be in the driving seat after Kai Havertz lobbed Bart Verbruggen to send the hosts into the break with a slender advantage.
However, the complexion of the contest completely changed when Rice was shown the first red card of his career following two bookable offences.
The England international received his second yellow for kicking the ball away, but eyebrows were raised that Joel Veltman escaped punishment for kicking through Rice in the same incident.
Fresh from his late winner against Manchester United, Joao Pedro sent the Brighton support into raptures when he pounced on a rebound to guide home the all-important equaliser.
Both teams threatened to score a winner with Havertz and Bukayo Saka forcing Verbruggen into action, while Raya made a desperate top to deny Jan Paul van Hecke.
The two sides ultimately had to settle for a share of the spoils, ending Arsenal and Brighton's 100% start to the new season.
As a result of the draw, Mikel Arteta's side have dropped points at home for just the second time in 2024, while Brighton have now avoided in four of their last five away games against Arsenal.
ARSENAL VS. BRIGHTON HIGHLIGHTS
Havertz goal vs. Brighton (38th min, Arsenal 1-0 Brighton)
Arsenal find the breakthrough. Saka gets to the ball ahead of Lewis Dunk before playing a perfectly weighted through pass to Havertz. The German calmly lifts the ball over the onrushing Verbruggen to net his second goal of the season.
Rice red card vs. Brighton (49th min, Arsenal 1-0 Brighton)
The hosts are down to 10 men. Rice gives away a soft free-kick on Veltman before he flicks the ball away. The midfielder is shown a second yellow for delaying the restart. Veltman is perhaps fortunate not to follow Rice down the tunnel after kicking through the England international.
Pedro scores vs. Arsenal (58th min, Arsenal 1-1 Brighton)
The Seagulls waste little time in making full use of the numerical advantage. Minteh is sent through on goal by Dunk's defence-splitting pass. The summer signing is denied by Raya, but Pedro is on hand to fire home the rebound.
MAN OF THE MATCH - JOAO PEDRO
Pedro's finish may have been a relatively simple one but like a true goalscorer, he was in the right place at the right time to net his second goal of the season.
In addition to his crucial equaliser, the Brazilian represented Brighton's main attacking outlet and always looked like he could trouble the Arsenal backline at any given moment.
While Pedro is our Man of the Match, Saka was certainly under consideration after providing an assist and posing a constant threat throughout the contest.
ARSENAL VS. BRIGHTON MATCH STATS
Possession: Arsenal 36%-64% Brighton
Shots: Arsenal 11-22 Brighton
Shots on target: Arsenal 7-4 Brighton
Corners: Arsenal 3-7 Brighton
Fouls: Arsenal 12-7 Brighton
BEST STATS
WHAT NEXT?
The Premier League will now pause for the international break before Arsenal and Brighton return to action in a couple of weeks.
The Gunners will face North London London rivals Tottenham Hotspur on September 15, while the Seagulls will play host to newly-promoted Ipswich Town on September 14.