Rose, 19, smashed home Manuel Almunia's looping fisted clearance on the full from 30 yards out after just 10 minutes.
The thunderbolt return screamed narrowly over Almunia's head and, though the Spaniard appeared to react slowly, it was hit with such venom that he had little chance.
Spurs doubled their lead through Gareth Bale a minute after the interval and, although Nicklas Bendtner pulled one back six minutes from time, the hosts held on to seal their first league triumph over their north London rivals since 1999.
Just as importantly, victory propelled them right back into contention for the fourth Champions League place and they now trail fourth-placed Manchester City by one point with five games remaining.
It was just the response Harry Redknapp was hoping for after Sunday's bitter FA Cup semi-final defeat by Portsmouth - a match that had taken extra time to settle.
There was little sign of tired legs, though Spurs - lifted by the return of skipper Ledley King from a thigh injury - rode their luck at times with Bendtner missing one good early chance.
Running out of ideas after having the better of the first-half possession, the Gunners wilted after the interval with Bale's strike appearing to be an insurmountable setback.
They found a second wind - inspired mainly by the 68th-minute arrival of Robin van Persie with Bendtner supplying their goal - but were frustrated by a string of fine saves from Heurelho Gomes.
The six-point gap with leaders Chelsea remains and, with four matches left, Arsenal have little chance of ending their five-year wait for silverware this season.
Much of the pre-match focus was on Sol Campbell's return to White Hart Lane and his every touch of the ball were predictably greeted with a chorus of jeers
But the former England defender almost silenced the home fans with a minute on the clock only for his goalbound header to be cleared off the line by Benoit Assou-Ekotto.
Michael Dawson made a crucial tackle on Tomas Rosicky as he was pulling the trigger from 10 yards out amid a strong start from the Gunners.
The promise ended, however, when Rose turned play on its head by driving home his contender for goal of the season.
Moments earlier Spurs had seen Thomas Vermaelen make a vital block of Roman Pavlyuchenko's close-range shot and Arsenal's problems deepened when the Belgian defender limped off to be replaced by Mikael Silvestre.
The goal had knocked the wind out of Arsenal's sails but slowly they clawed their way back into the match.
Bendtner was superbly teed up by Rosicky but, despite only being 12 yards out and having time on the ball, he still scuffed a poor shot towards Heurelho Gomes.
Panic reigned in Arsenal's defence when Silvestre gave Luka Modric a glimpse of goal with a careless touch but Almunia blocked and then Campbell came to the rescue.
Spurs replaced Rose with David Bentley at half-time and it took just a minute for the Champions League hopefuls to extend their lead.
Jermain Defoe did well to hold off Emmanuel Eboue and pick out Bale with a beautifully-weighted pass.
Silvestre's lack of pace was exposed as the in-form Bale raced inside him and planted the ball beyond Almunia.
Defoe and Pavlyuchenko were then scrambling frantically to take advantage of hesitancy in the visitors' defence as Spurs pushed for a third.
Bendtner dragged a shot narrowly wide of the left post and, in the 68th minute, the Dane was joined up front by Van Persie.
Van Persie forced a fine reflex save from Gomes and the Brazilian then punched clear three successive hanging balls amid a mass of bodies.
Better was to come from Gomes, however, as he made a magnificent stop from Van Persie's perfectly placed free-kick before then diving to keep out Campbell.
But he was powerless to prevent Arsenal pulling one back with six minutes to go as Spurs' defence failed to pick up Theo Walcott's pass which Bendtner slid across to bundle into the net.
The Gunners poured forward in search of the equaliser but Spurs held out for a precious win over their north London rivals.