Two-time major champion Azarenka reached the Miami semi-finals for a fifth time after overcoming Putintseva – and a 45-minute stoppage due to a power outage.
The delay in the first set was caused by the power going out on the electronic line-calling system and when the match resumed, 34-year-old Azarenka saved three set points before clinching the tie-break.
Unseeded Kazakh Putintseva dominated the second set, but dropped serve twice in the third to allow Azarenka, who defeated Britain’s Katie Boulter in the fourth round, to reach a first semi-final in Miami since 2018.
Rybakina, who is the highest ranked women’s player remaining in the tournament, lost in last year’s final to the Czech Republic’s Petra Kvitova.
The 2022 Wimbledon champion had two match points in the second set on Tuesday but Sakkari fought back and won the tiebreaker to force a decider.
A single break of serve in the third set was enough for Rybakina to wrap up the victory after a gruelling two hours and 48 minutes.
“I have no words because I’m so tired but I’m really happy with the win,” Rybakina, 24, said.