After not dropping a set in her opening two matches at the WTA International hard-court event, the Slovakian was initially surprised by her Kazakh opponent on Friday, but was able to recover well and secure a 4-6 6-2 6-4 victory.
Cibulkova took control in the second set by winning five consecutive games and wrapped up victory in a topsy-turvy decider.
Only three games went with serve in the final set and it was this year's Australian Open runner-up who eventually prevailed, sealing a last-four berth on her second match point.
Cibulkova will face defending champion Karolina Pliskova after the latter swept past Cagla Buyukakcay 6-4 6-4 in 80 minutes.
Pliskova seized the advantage with an early break and, after failing to take her first set point in the ninth game, moved ahead by holding serve to love.
The Czech then converted two early breaks in the second to claim a 4-1 lead, which proved decisive even though Buyukakcay pulled a break back.
The second semi-final will see Donna Vekic play Shuai Zhang after both completed quarter-final victories on Friday.
Zhang looked in complete control of her match with Magda Linette after taking the opening set in 43 minutes, but was ultimately made to battle for a 6-2 2-6 6-1 win.
Linette levelled the match with two breaks of serve in the second set, before Zhang recovered from losing her serve at the start of the third to win six straight games.
Vekic's progress was less problematic as she beat Patricia Mayr-Achleitner 6-4 7-5.
The 17-year-old saved nine break points during the match, and was able to convert four of the eight opportunities she created.