Rain the previous day had interrupted her battle with Pole Radwanska but when she returned to court in sunshine, trailing by a set, she blazed to a 4-6 7-6(2) 6-3 victory.
Puerto Rico's Puig had the benefit of a night's rest, having taken care of Kristina Mladenovic the previous day.
But she was no still match for the attacking play of Cibulkova who cruised to 6-2 6-1 victory.
Radwanska had looked set to reach the semis, but never got going after the break.
"Never know how it's gonna happen if we finished that match yesterday. But definitely different conditions today," she said.
Cibulkova will contest her fourth final of the year, and the first of her career on grass, on Saturday against 10th seeded Czech Karolina Pliskova who beat Britain's Johanna Konta 6-7(5) 6-3 6-3.
"When we started to practice after Roland Garros, I said to my coach, it's shame that the clay is over," Cibulkova, who will be seeded 20th at Wimbledon told reporters.
"I said: 'I want to play more on clay. I don't want to change the game and go on the grass, right?' and he said: 'You know, Domi, you'll be fine. You're playing good. You will just change small things in your game. You will see you will be fine on the grass.'"
"I'm happy he was right."
Konta, the first British woman to be ranked in the world's top 20 since 1986, had earlier beaten Russian Ekaterina Makarova 7-6(5) 6-4 in another delayed quarter-final.
(Reporting by Martyn Herman; Editing by Richard Balmforth)
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