Saturday's semi-finals will see Bertens take on White Group victor Elina Svitolina of Ukraine, before Stephens faces big-hitting Czech Karolina Pliskova in the night match of a tournament that has seen the four lowest-ranked players advance.
However, the second set - which lasted 48 minutes - was a different story as number six seed Svitolina finished strongly, winning three games in a row to take it 6-2 and square the match.
And Stephens was the final woman to notch her place in the semi when she secured her fifth straight victory over top seed Angelique Kerber.
The Ukrainian nearly missed qualifying for the WTA Finals after some early-round losses in the lead-up to Singapore. "I think she's still playing similar [to last year], but maybe she was missing much more before", Pliskova said in press after their match earlier in the week. I think I should be doing well on all surfaces.
But Stephens clawed her way back and screamed when Pliskova's forehand went wide to give her a first game after just over 50 minutes of play.
Svitolina entered the final trailing Stephens 1-2 in their career head-to-head encounters.
"It was just getting worse and worse every time I played", said Osaka.
Osaka was in tears after her retirement, which left her without a win at this tournament.
The Ukrainian broke to start the final set, but Bertens hit right back breaking in an epic second game which saw eight deuces and four break points for the Dutchwoman. Halep finally won a major, but the effort cost her dearly: The 5-foot-6 Romanian dynamo has a herniated disc.
That error threatened to prove costly when Kerber broke serve to keep the set alive, but the left-hander promptly lost her own serve for a third time to gift Stephens the set. "Even though my results were not good since Wimbledon, I had four or five weeks where I was on court and in the gym working the day after a loss", she added.
The 21-year-old learned a lot during her debut week in Singapore.
Svitolina heard the jibes, and she carried a chip on her shoulder.
"I knew it was going to be a battle and just tried to play as hard as I could". I think tomorrow will be a great match.
Humiliated and embarrassed by her performance, the 2017 U.S. Open champion roared with relief when she prevented the whitewash and then, remarkably, carried the momentum on to a 0-6 6-4 6-1 victory and a Sunday showdown against Elina Svitolina.