The Gunners took the lead when Aarson Ramsey back-heeled the ball past Spurs keeper Lloris just before half-time.
Arsenal midfielder Coquelin shot his side in the foot after the interval when he was sent off for a second bookable offence.
Tottenham took advantage of Arsenal going down to 10 men and scored twice in quick succession thanks to goals from Alderweireld and Harry Kane.
Arsenal have been criticised for lacking fighting spirit over recent weeks but managed to get themselves back into the game when Alexis Sanchez found the back of the net on 75 minutes.
The draw saw the Gunners remain in third place on 52 points while Tottenham stayed in second on 55 points. Leaders Leicester City could move five points clear of Spurs with a victory over Watford today.
Tottenham v Arsenal
Stadium: White Hart Lane
Kick-Off: 12:45 GMT
Tottenham confirmed XI: Lloris; Walker, Alderweireld, Wimmer, Rose; Dier, Dembele; Lamela, Alli, Eriksen; Kane.
Subs: Vorm, Trippier, Davies, Mason, Carroll, Son Heung-min, Chadli
Arsenal confirmed XI: Ospina; Bellerin, Mertesacker, Gabriel, Gibbs; Coquelin, Ramsey; Elneny, Ozil, Sanchez; Welbeck.
Subs: Macey, Chambers, Monreal, Giroud, Campbell, Flamini, Walcott
Manager: Arsene Wenger
Goals: 39′ Ramsey, 60′ Alderweireld, 62′ Kane, 76′ Sanchez