Swiatek defeated Diana Shnaider 6-0 6-7 (3-7) 6-4 to stay on track to defend her title in the Spanish capital.
"I liked yesterday. I think the whole world was just more relaxed," she said. "I enjoyed the time off, and I tried to just recover, because we need to savour these free days."
Swiatek said she took away food from the tournament site and relaxed at the hotel until the power came back on.
"I just chilled and used that time to not think about what I should do," she said. "There was no signal, so basically no one used their phones. It was nice, I liked it."
Most parts of Madrid regained power overnight, but the Caja Magica was still without electricity early Tuesday, causing a delay in the opening of the gates for fans.
The power came back quickly, and organisers did not have to alter the day's schedule of matches, although there was the possibility of the night session being delayed with 22 matches being held over from Monday.
Swiatek cruised in the first set but had to save 11 of 13 break points to seal a hard-fought victory against the 13th-seeded Shnaider.
Swiatek hasn't gone past the semi-finals since Roland Garros last year. She will next face Madison Keys, who defeated Donna Vekic 6-2 6-3.
It will be Swiatek's first meeting with Keys since she lost, despite having match point, in the semi-finals of the Australian Open that was won by Keys.
Top seed Aryna Sabalenka beat Peyton Stearns 6-2 6-4 and will meet Marta Kostyuk who defeated Anastasia Potapova 6-3 6-2.
In the same half of the draw Elina Svitolina beat Maria Sakkari 6-3 6-4 and will meet Moyuka Uchijima, who defeated Ekatarina Alexandrova 6-4 7-6 (7-5).
The other quarter-final will be between Coco Gauff and Mirra Andreeva who competed their last-16 ties before the blackout began on Monday.
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