Summer signing Timo Werner's scored in the 19th minute when the former RB Leipzig man settled a cross from the right-handed side and fired through a thicket of legs to beat a helpless Hugo Lloris in goal.
Chelsea's new goalkeeper Edouard Mendy had a confident display in his first game since switching from Rennes, making a good save to keep out a vicious second-half shot from the hosts.
Other than that, Spurs looked listless for much of the second half until a Sergio Reguilon cross found Erik Lamela at the far post and he tapped home to level the score and set up the penalty shootout.
The match was marked by a pair of bizarre scenes with Jose Mourinho and Frank Lampard involved in a shouting match on the sidelines in the first half and Eric Dier leaving the pitch late in the second half to apparently use the bathroom.
FC panel react to Mourinho chasing Eric Dier to the toilet
The FC crew can't figure why Jose Mourinho had to chase after Eric Dier when Dier needed a bathroom break.
"I'm so happy, I'm proud of how we played apart from the first 15 minutes," Dier, who played on Sunday in the Premier League, said after the match. "We showed them too much respect but then we were fantastic.
"After playing on Sunday it wasn't easy, maybe that's the reason I had to run off but we got through it! All of the staff and the players put the effort in.
"Jose wasn't happy, but there was nothing I could do about it, nature was calling. I heard there was a chance while I was off the pitch but thankfully they didn't score."
The result was a bit of payback for Spurs after Chelsea beat them in a shootout in the 2018-19 Carabao Cup semifinals.