Gabriel strikes as Arsenal hit wasteful Spurs with sucker punch
Mikel Arteta's side were without skipper Odegaard due to an ankle injury suffered on Norway duty, while Rice served a suspension for his red card against Brighton.
Losing two such influential players appeared to make Arsenal vulnerable on their trip up the Seven Sisters Road to face their bitter rivals.
But the Premier League title contenders dug deep with a combative display, benefitting from Tottenham's latest display of wasteful finishing before Gabriel bagged the winner in the second half.
The Brazilian defender's first goal since February made it three wins from four league games for unbeaten Arsenal as they chase a first title since 2004 after two successive runners-up finishes.
Arsenal have now won on their last three visits to Tottenham, losing just once, in May 2022, in their last eight meetings with their hated neighbours.
The second placed Gunners are two points behind Manchester City as they turn their attention to their Champions League opener at Atalanta on Thursday before travelling to face Pep Guardiola's champions next weekend.
The 196th north London derby showed Arsenal have steel and spirit aplenty, but they also needed a dose of good fortune as Tottenham let them off the hook in the first half.
Failing to turn dominant spells into goals has been a familiar story for Tottenham this season and once again Ange Postecoglou's men were let down by their lack of cutting edge.
Tottenham's second successive defeat leaves them with only one win from four games in a worrying start to the season for Postecoglou after their late swoon last term cost them a top four place.
Turning to Jorginho and Gabriel Martinelli to replace Rice and Odegaard, Arteta had urged his team to show the "resilience, depth and hunger" needed to overcome their selection issues.
Set-piece specialists
Arsenal had to call on all those qualities during a vibrant start from Tottenham, with David Raya making a fine save at his near post after Son Heung-min's pass picked out Dejan Kulusevski for an instinctive strike.
Raya had to make an even more agile stop when Kulusevski's cross flashed through a crowd of players toward the far corner until the Spaniard clawed it away at full stretch.
William Saliba had to make a crucial, last-ditch tackle to deny the unmarked Dominic Solanke, who took too long to shoot inside the area after Son pressured Arsenal into conceding possession.
Tottenham's profligacy gave Arsenal encouragement and Kai Havertz rose highest for a close-range header that forced a superb save from Guglielmo Vicario, who was also equal to Martinelli's curler moments late.
Picked out by James Maddison's cross, Solanke did well to hold off Gabriel but his looping header dropped narrowly wide.
Amid a flurry of aggressive challenges, the simmering derby tension boiled over after Jurrien Timber's tackle on Pedro Porro left the Tottenham right-back writhing on the turf.
Vicario confronted the Arsenal defender, with both players booked after a melee featuring a plastic bottle thrown from the stands.
By the time Kulusevski crudely up-ended Jorginho on the stroke of half-time, seven players had been booked in a fractious clash, including five from Tottenham.
Micky van de Ven's glancing tested Raya soon after the interval, but toothless Tottenham were hit with a brutal sucker punch in the 64th minute.
With 23 goals from set-pieces since the start of last season, Arsenal have become dead-ball specialists this was another expertly worked routine.
Saka's in-swinging corner arrowed towards Gabriel, who got away with a push on Cristian Romero before powering his header past Vicario from five yards.
Tottenham penned Arsenal back in the closing stages, but couldn't find a way through as Kulusevski blasted just over.