The German side conceded two late goals before clinging on to secure a 4-3 aggregate win and book their place in the final at Wembley.
"It seems like we only can do it with drama," Watzke said on the club's official website.
"For the first time in my life I had to give up due to heart problems. I went to the toilet for the last minutes, locked myself in, covered my ears and looked at my watch. I had all kinds of thoughts going through my head."
Director of sport Michael Zorc also admitted he struggled to cope with the nailbiting finish, and five minutes of injury time, after goals from Karim Benzema and Sergio Ramos put the hosts within sight of a remarkable fightback.
"We have earned ourselves a huge reputation in Europe. In the last minutes I was close to a heart attack. But when there is a positive outcome, it's all good.
"It was great. In the second half we played a really good game. In the space of 10 minutes we had four golden chances.
"When you don't take them then you are often punished. That's why it was really tight again. That was not in the plan. But, in retrospect, it was only a slight deviation from the plan."
Defender Neven Subotic was one of Dortmund's star performers on the night, along with central defensive partner Mats Hummels and goalkeeper Roman Weidenfeller.
Subotic said: "It was 98 minutes of hard work. I can't remember the last time I faced such a challenge. Both teams wanted to be in the final.
"After we went 2-0 down I thought, 'When you clear the ball, then hoof it as far away as you can'.
"When I saw it was five minutes added time I thought I would die.
"But we played (the match) down well in the last three or four minutes. The most dangerous phase was the four minutes before stoppage time."
The only negative for Dortmund was an injury to attacking midfielder Mario Gotze, who had to go off after 14 minutes with a hamstring tear.
That could leave the 20-year-old, who will join Bayern Munich in the summer, facing a battle to be fit for the final on May 25.
"It's going to be race against time for the things to come," Dortmund coach Jurgen Klopp said.