The Spanish defender had an eventful home debut for Thomas Tuchel's side, having joined from Brighton and Hove Albion at the start of the season.
Cucurella set up the opening goal of the game when his 19th minute corner found the boot of Kalidou Koulibaly to lash home.
Chelsea held a 2-1 lead until second half stoppage time when Spurs defender Cristian Romero pulled Cucurella down by his hair but referee Anthony Taylor and his VAR team took no action and the visitors scored from the ensuing corner to nick a point.
The incident was another sideshow in a game that saw opposing managers Thomas Tuchel and Antonio Conte sent off after the final whistle over a post-match handshake gone wrong.
Cucurella has opened up on the incident but has refused to alter his hairstyle because of his opponents' antics.
“No, no [I will never get my hair cut], this is my style,” the 24-year-old said at Cobham on Wednesday.
“For me, it is no problem. Before the game, I spoke with him [Romero] and it is a fight on the pitch. I have no problem with him and it is only an action in the game.
“Maybe it was an important action as it maybe changed the final result. I think we will have another opportunity in a few months and this is our opportunity.
"For me, yes [it is a foul] because it was clear. In the game, out on the pitch, I didn't see the video. But I think with the video it is very clear. This is maybe one error to the referee or the VAR. But sometimes the referees make good decisions, sometimes not good decisions.
“This is not my problem. The players are focused on the game and maybe we have to work hard to not concede this goal in the last minute in the next games.
“The referee in the stadium said they were checking a red card and for me, it's very clear. The video is very clear. At this moment, with the referee, I don't know what happened.
“Sometimes it's good, sometimes it's bad. This is football and in this game, it was an error. But in another game, the referee takes a good decision.”
