Kane, who was playing in his first derby match for his boyhood club, scored his 21st and 22nd goals of an incredible breakout season after Tottenham fell behind controversially in the first half.
Coach Arsene Wenger had raised the possibility that key striker Alexis Sanchez might recover from a hamstring injury in time.
Sanchez didn't make it but former Real Madrid star Mesut Ozil, himself only recently returned from injury, skillfully fired home after Olivier Giroud had fired the ball across the penalty box.
Replays showed that Ozil was marginally offside.
The setback didn't trouble Tottenham for long, who dominated for long periods as Arsenal conceded possession and played counter attacking football.
Kane leveled the match after Arsenal goalkeeper David Ospina could only manage to parry a high ball in to the striker's path.
His second, however, was unstoppable. Kane rose highest in the penalty box to cleverly steer the ball back across goal and into the top left hand corner.
"I'm enjoying it. I just want to be out on the pitch playing with my mates," said an emotional Kane, who had tears in eyes following a standing ovation as he left the pitch.
"I don't think I can put into words," he added. "It's my first London derby. It's incredible."
The victory saw Mauricio Pochettino's team leapfrog Arsenal in to fourth place.
Wenger, meanwhile, was unhappy with his team's performance but also had words of praise for Kane, who is the leading English goalscorer in the Premier League.
"If you don't give him a game quickly, somebody else will give him a passport," he said of calls for Kane to be called up to the England national team squad.
Elsewhere on Saturday afternoon, champions Manchester City fell further behind Chelsea in the Premier League title race after being held 1-1 by relegation threatened Hull City.
Hull took a shock lead at the Etihad Stadium, where Manchester City had been impenetrable until recently, when David Meyler fired home.
The goal looked like it would secure Hull City a vital three points, until England international James Milner scored a late, late equalizer deep into injury time.
Whilst Manchester City will no doubt be happy to have at least salvaged a point, they will be less happy to have seen Chelsea pull seven points clear at the top of the table.
Chelsea coach Jose Mourinho had never been in charge of a winning side away from home against Aston Villa. But an early Eden Hazard goal looked to have set Chelsea up for an easy victory.
Aston Villa had only scored eleven times this season in the league yet managed to pull level through Jores Okore before Branislav Ivanovic restored Chelsea's advantage, as well as tightening Mourniho's grip on another Premier League title.
In Saturday's late game Liverpool drew 0-0 with Everton in the Merseyside derby.