Arsenal substitutes Stephan Lichtsteiner and Aaron Ramsey took exception to Dier’s taunting, prompting a bout of pushing and shoving.
Son Heung-Min’s pace was too much for Arsenal to handle and Rob Holding responded by making a rash sliding challenge on the South Korean four minutes later.
Kane lashed home the resulting penalty for a record-equalling eighth Premier League goal in north London derbies.
Arsenal’s sense of injustice spurred them on and Emery responded by sending on Ramsey and Lacazette, back from a groin injury, at half-time.
They proved inspired moves as Gabon forward Aubameyang conjured a majestic equaliser in the 56th minute.
Tottenham had reverted to their earlier timid performance and Arsenal went ahead in the 74th minute when Lacazette’s shot deflected in off Dier.
Torreira delivered the knockout blow three minutes later and there was still time for Vertonghen to see red in the 85th minute.