Barty, who won the tournament’s most recent title in 2019, trailed 5-2 in the final set. In the next game she faced a match point, which she saved by ripping a weak serve for a winner.
Barty fell behind 0-40 serving in the final game but again rallied against Kucova, a Slovakian ranked 149th. Barty closed out the victory with a service winner, and then tapped her temple with her index finger, as if in tribute to the way she kept her composure in the crunch.
Playing away from her native Australia for the first time in more than a year, Barty won despite an unreliable forehand. She whacked 40 unforced errors on that side.
On a warm, humid afternoon, heat breaks were imposed for some matches, and players slid bags of ice under their caps during changeovers to cool off. The conditions were too much for Brit Jack Draper, who was overcome by the heat, collapsed on set point and retired against Mikhail Kukushkin trailing 7-5.
Wild card Ana Konjuh of Croatia upset No. 18-seeded Madison Keys in the second round, 6-4, 6-2. Konjuh’s career has been slowed by four elbow surgeries, and in the past two days she has earned her first two wins in main-draw WTA play since 2018.
Kokkinakis is ranked No. 243 and mounting a comeback from a shoulder injury. He beat Roger Federer as a qualifier in Miami three years ago and returned to the tour last month after being sidelined more than a year.