The Spanish world number two overcame Greek Tsitsipas 6-4 4-6 6-2 in London.
Nadal, 34, had been the more consistent player until losing serve to drop the second set, but fought back to dominate the decider.
The 20-time Grand Slam champion will play Russian Daniil Medvedev in the semi-finals at the O2 Arena on Saturday.
Medvedev is already guaranteed to win Group Tokyo 1970 with Novak Djokovic and Alexander Zverev contesting the remaining last-four place at 14:00 GMT on Friday.
“I was playing, I think, a great level of tennis for almost two sets,” said Nadal, who hit 32 winners and made just 13 unforced errors.
“The score was close, but in terms of feeling I felt that I was playing better than him. Then the situation changed. He won that game and then I was starting to suffer a little bit more with my serve.
“But it’s always a good feeling when you are playing well and you’re playing against the best players of the world. You know you’re going to be in a tough fight. That’s what happened tonight. I was able to find a way to win. Very happy.”