Tottenham twice trailed to Cristian Romero's own goal and Bukayo Saka's penalty at the Emirates Stadium.
But Tottenham captain Son led the fightback on each occasion with a first-half leveller and then another equaliser after the interval to extend his side's unbeaten start to the Premier League campaign to six games.
"It's a derby, it's always a tough game. I think the performance was fantastic. We gave everything until the final whistle, so the fans can be proud of the game," Son said.
"We wanted to win so one point is a shame, but the performance was perfect."
Ange Postecoglou's team conceded a second-half penalty when Romero was ruled to have deliberately blocked Ben White's shot with his hand.
Despite that blow, Tottenham showed they are made of sterner stuff than in recent seasons as they immediately recovered to equalise through Son's cool finish from James Maddison's pass.
Although they failed to end their 13-year wait for a league win at Arsenal, fourth-placed Tottenham remain above the fifth-placed Gunners on goal difference.
"Look the referee made a decision, we can't just argue about this decision. We have VAR in this situation," said Son, who now has five goals this season.
"Obviously the derby is an emotional game. It's a decision we accepted and we bounced back really strongly.
"Coming back at the time was important. We showed great character and Maddison showed great bravery."
