Aaron Ramsey delicately gave Arsenal the lead against the run of play in the first half, but a red card for Francis Coquelin for a second bookable offence just after the interval turned the game on its head.
Spurs scored twice in two minutes through Toby Alderweireld and Harry Kane to move into a 2-1 lead, but Alexis Sanchez's 76th-minute strike levelled it up and earned the Gunners a point.
In a truly exhilarating first half, the hosts were doing all of the early pressing. Danny Rose got forward from left-back and caused Hector Bellerin endless problems, while Mousa Dembele prevented the Gunners from playing their free-flowing football.
However, the likes of Kane and Christian Eriksen were limited to shots from long range, with Per Mertesacker and Gabriel Paulista standing strong in the heart of the Arsenal back line.
Tottenham did come perilously close to breaking the visitors down just before the half-hour mark, though. Kyle Walker's driven cross was turned goalward by Erik Lamela, but David Ospina was there to make a flying save and keep the score 0-0.
The Colombian was under a lot of pressure to fill the sizeable gloves of the injured Petr Cech, but Paddy Power jested the save from Lamela was one the Czech star would have been proud of:
Spurs continued to pile the pressure on Ospina's goal, yet the Gunners always looked a threat on the counter-attack. Danny Welbeck made some excellent runs and kept defenders busy throughout, and it was no surprise to see the Englishman at the heart of the opening goal.
The ball broke to Welbeck on the left flank following a throw-in, and he had the presence of mind to pick out the rampaging Bellerin, who squared across the box for Ramsey to skilfully flick home.
BBC Sport's Phil McNulty was one of many who praised the Welshman's finish, which left goalkeeper Hugo Lloris with no chance:
The goal stunned both the White Hart Lane faithful and Mauricio Pochettino's players, as Arsenal suddenly looked a different outfit going forward.
Welbeck could have doubled the visitors' lead soon after when Alexis Sanchez picked the striker out in the box, but he just couldn't get enough purchase on the header.
Arsene Wenger would have been the much happier manager at half-time, and if his side could have stayed disciplined and stuck to his game plan, three points were there to take back to the Emirates Stadium.
Just 10 minutes into the second half, though, Coquelin had a moment of madness. The Frenchman flew in on Kane on the left-hand side having already been booked. Referee Michael Oliver had no choice but to give him his marching orders.
The Mail on Sunday's Oliver Holt commented on Coquelin's challenge:
Five minutes later, Alderweireld fired Tottenham level.
An Eriksen corner was met by Lamela, and the ball broke kindly for the Belgian, who fired home emphatically, as the Times' Henry Winter noted:
It got an awful lot worse before it got better for Arsenal. Two minutes after Alderweireld's equaliser, Kane produced one of the goals of the season to send White Hart Lane into raptures.
Dele Alli did well to keep the ball in play and feed Kane on the left, and the England man curled a brilliant shot past the despairing Ospina. Gary Lineker sung his compatriot's praises on Twitter:
Tottenham went in search of a third goal to put the game to bed, but at 2-1, Arsenal were always in the match.
Attacks were few and far between for Wenger's men, but one venture forward was all it took for them to draw level.
Bellerin found space on the right flank, passed the ball into Sanchez 12 yards out and the Chilean's right-footed finish just evaded Lloris in the Tottenham goal. It was his first goal since October, according to Sky Sports Statto:
The game cried out for a winner, and both sides could have nicked all three points late on. A brilliant Sanchez free-kick was saved well by Lloris, while Gabriel almost turned through his own net after swiping at an Alli cross.
Ramsey had an opportunity to double his tally for the day in stoppage time, but Kevin Wimmer's excellent challenge saved Spurs.
Tottenham closed the gap on Leicester City at the top of the league to two points, while Arsenal remained three points adrift of their rivals.
The Gunners have been criticised for not showing enough heart of late, but the way they conducted themselves with 10 men was impressive. Arsenal host West Bromwich Albion in their next league game. Spurs travel to bottom club Aston Villa.
Kane told BBC Sport his side should have killed the game off while they had the chance and conceded he never thought the game was won:
We are disappointed. It's an opportunity missed. We should have finished the game off. We should have got the third and fourth, and that would have been the end. But it if you give top sides room, they punish you.
I didn't think my goal had won it; it put us in a very good position. Maybe we dropped off and gave them too much time on the ball, but it's all part of the learning curve. There is a long way to go, and we have to take each game as it comes.
Wenger, meanwhile, praised Arsenal's character, though he lamented the lack of discipline that almost cost the Gunners dearly, per BT Sport (via BBC Sport):
I am proud of the spirit, but there are big regrets. Going down to 10 men was the big regret for the day, and it was hard to take. We told Francis Coquelin at half-time because he had yellow already, but Eric Dier deserved a second yellow as well, and that could have changed the game.
I never questioned the character of the team. When you lose, you have to respond on the pitch, and we did that in a convincing way.
That being said, the Gunners supporters will be upset with more dropped points at a critical juncture in the campaign. Wenger touched on pressure from the stands, per Fox Australia's Daniel Garb:
Arsenal are without a win in their last five outings and have conceded nine goal in their last four games. Wenger is used to an intense spotlight shining on his side approaching the end-of-season run-in, and the manager can only rue more points gone astray.