Swiatek has reached the last four in each of her past three appearances at the event in California and will take on Ukrainian Marta Kostyuk on Friday.
Kostyuk, the 31st seed, secured a first WTA 1,000 semi-final appearance with a 6-0 7-5 win over Russian 28th seed Anastasia Potapova.
Wozniacki, 33, was enjoying her deepest run at a tournament since returning to the circuit last season having come out of retirement after giving birth to her two children.
“Huge respect for her, and I’m sad that we had to finish that way,” Swiatek, 22, said after the match.
US Open champion Gauff, 20, trailed 3-1 in the second set and was down a break point but she rallied to reach the semi-finals at Indian Wells for the first time, becoming the youngest American since Serena Williams in 2001 to reach the tournament’s last four.
“I’m really happy to be through today’s match,” Gauff said.
“It wasn’t the best style of the serving, but overall I think when it comes to the groundstroke game and returns, everything else, honestly, I played well. I just try and take the positives into the next match.”
Sakkari, 28, is next up for Gauff after the Greek ninth seed beat American Emma Navarro 5-7 6-2 6-4 to reach the semi-finals for a third consecutive year.