German top seed outlasted US Open champion Osaka, before American fifth seed overcame a mid-match slump to defeat Dutchwoman Bertens
Sloane Stephens and Angelique Kerber win gruelling matches as WTA Finals group goes down to the wire
It remains all to play for in the WTA Finals Red Group after two fiercely-contested encounters in Singapore on Wednesday.
In the first match, top seed Angelique Kerber produced a dogged display of defensive tennis to tame big-hitting Naomi Osaka 6-4, 5-7, 6-4, before American Sloane Stephens recovered from a mid-match slump to defeat Kiki Berttens of the Netherlands 7-6, 2-6, 6-3.
Kerber's victory evened up her record at 1-1 in round robin play as Osaka slumped to her second straight defeat, leaving the US Open champion needing to beat Bertens in her final group match to have any hope of advancing.
The early match began tentatively with three consecutive breaks of serve falling in the German's favour as Osaka looked to dictate with her power, while Kerber was content to counter-punch off the back foot and keep the ball in play.
Osaka fashioned five break points when Kerber served for the set at 5-4 but the German was able to stave off all of them and sealed the opener with a booming serve.
Osaka's inability to make the most of her potent serve looked to have cost her the match when she was broken to love in the ninth game of the second set but she then reeled off three games in a row to level the contest.
The third set was a war of attrition with both players stretching every sinew to hold serve until Osaka cracked first in the seventh game, gifting Kerber the crucial break.
The break was enough for Kerber to seal a brilliant win, with Osaka crushing a backhand into the net on the German's first match point.
"I had my chances in the second set but she played some unbelievable tennis to win that one, and it felt like there was only one or two points in the decider," Wimbledon champion Kerber said in a post-match interview.
"It was great to win such a great match and to be able to look forward to another one in two days from now."
The second match was a slow burner, with the crowd taking time to warm to the contest, as Stephens raced into a 3-0 lead before Bertens dragged herself back.
Improved serving from both players ensured the opener went to a tiebreak and it was Stephens who edged ahead on her third set point when Bertens sent a forehand into the tramlines.
Bertens capitalised on a drop in intensity from Stephens to reel off four games in a row on her way to taking the second set.
The Dutchwoman carried that momentum into the start of the decider, breaking to move 2-0 ahead, but Stephens finally woke from her slumber to reassert her authority and close out a memorable victory.
"I just tried my best from the very first point of the final set. It didn't go my way immediately but I just kept battling and I'm really pleased to pull through," Stephens said.