"It was a beautiful match, one of those the fans show up to watch," Ter Stegen told Movistar Plus.
"I’m very proud of myself. I think since last year I’ve been going through a great moment in my career and in my life.
"It was a very difficult match mainly after the break because they dominated possession and made it really though for us. They were superior in the midfield, had control of the match and we had to fight really hard to advance."
Barcelona started well but once Pedri's 23rd minute goal was ruled out by the VAR for an offside in the build-up, they started to lose ground to Betis.
After two outstanding reflex saves by Ter Stegen blocked Luiz Henrique and Rodri's bullet-like shots, Ousmane Dembele burst forward in a fine individual effort before he passed the ball to Lewandowski, who beat goalkeeper Claudio Bravo from close range to score the opener.
Lewandowski thought he had put Barca back in-front with a strike three minutes later but VAR again ruled-out the goal for an offside in the build-up and the match went into extra time.
Ansu Fati put Barcelona back in front, finishing off a stunning move from just inside the area, but Betis fought back again and equalised thanks to another fine effort from Luiz Henrique, whose cross was brilliantly finished by substitute Loren Moron with a heel-kick from just inside the six-yard-box.
Barcelona did not put a foot wrong in the shootout while Betis were left ruing the penalty misses by Juanmi and Carvalho.