Down a set and a double break, the Greek player looked on track for a crushing defeat at the hands of her American opponent.
But Sakkari rallied just at the crucial moment in the second set, breaking serve twice to level the scores at 4-4 before taking a third break to force a decider.
Collins opened the third set with a break, but Sakkari closed out the match in emphatic fashion, winning 3-6 7-5 6-2 after just under two hours and 15 minutes. She will now face Mihaela Buzarnescu in the final in San Jose.
The Romanian came back from a set down against Elise Mertens to reach the first Premier-level final of her career.
In a rematch of the pair's Hobart International final in January, Buzarnescu stepped up a gear in the second set, and reeled off six games in a row in the third to take a 4-6 6-3 6-1 victory.
At the Citi Open in Washington, number seven seed Donna Vekic will face former winner Svetlana Kuznetsova in the final.
Vekic was forced to complete two matches on Saturday after rain put a halt to her quarter-final meeting with Poland's Magda Linette on Friday.
The Croatian number one was 3-0 up in the first set when the match resumed, and she wasted no time in racing to a 6-1 7-6 (7/0) victory.
Just eight hours later, Vekic was back on court, facing China's Zheng Saisai in the final four.
Vekic rallied from a break down in the opening set, breaking her opponent twice and winning the last three games in a row to seal a one-set lead.
She continued her momentum in the second set, winning four of the last five games to clinch the match in just over 90 minutes.
Kuznetsova will prove a fierce opponent in the championship match, having won the title in 2014.
The Russian defeated Andrea Petkovic 6-2 6-2 to book her place in the final, having finished off her quarter-final against Yulia Putintseva earlier in the day.
Kuznetsova won six straight games in the first set against Petkovic after losing her serve early on, and then took two breaks in the second for a 6-2 6-2 win.
