It took the 27th-seeded Belarusian, who took home the trophy at this event in 2016, about three hours to defeat Putintseva 7-6 (7-4) 1-6 6-3.
Rybakina, ranked fourth, defeated eighth-seeded Maria Sakkari of Greece. 7-5 6-7 (4-7) 6-4.
On Wednesday, however, Frenchwoman Caroline Garcia, ranked 23rd, was defeated 6-3 6-2 by American Danielle Collins.
Collins, 30, a former runner-up at the Australian Open, intends to exit the tour at the end of the current campaign.
Since Ashleigh Barty defeated her at Melbourne Park, she only needs one more win to advance to her first final.
The Russian Ekaterina Alexandrova, who defeated world champion Iga Swiatek before falling to fifth-seeded Jessica Pegula, will be seen by Collins.
The 14th-seeded Alexandrova took the crucial break to lead 5-4 in the third set and went on to win the match and advance to her second WTA 1,000 event semifinal.
