The fifth seed fired his way through two comfortable sets. The turnaround that followed in the fourth set was hard to envisage.
Tsitsipas didn’t see it coming as the Greek star was visibly frustrated when the Australian mounted his comeback.
Kokkinakis found his first serve, produced some clutch points to escape pressure moments and forced another tie-breaker.
Tsitsipas was warned for receiving advice from his coaches’ box and fell behind 5-1 in the fourth set tie-break.
The Greek star worked his way back to 5-all but after a big serve from Kokkinakis, he produced his best shot of the afternoon.
The 24-year-old ripped a brilliant backhand down the line on set point, forcing a fifth and final set.
Tsitsipas broke the Australian midway through the decider and held his nerve to close the match.
Earlier, unheralded Australians Alexei Popyrin and Chris O’Connell were unable to extend runs at their home slams beyond the second round.
Popyrin’s clash with South African Lloyd Harris also went the distance.
Unfortunately for the Sydney product, there was no fairytale finish to mimic his career-best result against world No.15 David Goffin in the first round.
Popyrin had not pinched a set against Harris in either of their previous meetings but he broke that duck in the second set.
The Australian produced his best tennis in a thrilling fourth set tie-breaker to force a decider but ran out of steam in the fifth.
Popyrin had never won a five-setter before stunning Goffin on Tuesday and after that match, he predicted his body would be particularly sore in the next 48 hours.
Popyrin was on the money as his legs deserted him in the final set.
O’Connell had his best chance to feature in the third round of his home slam against journeyman Moldovan Radu Albot but the Sydneysider failed to fire.
He was comprehensively beaten 2-5, 5-7, 6-7 (10).
It’s just the second time in eight years on tour that Albot has progressed to the third round of a grand slam.
Away from the Australian men, there were no big upsets on a straightforward day at Melbourne Park.
Italian ninth seed Matteo Berrettini kept his hot run of form since arriving in Australia going with a 6-3, 6-2, 4-6, 6-3 win against qualifier Tomas Machac.
Fellow top 10 talent Andrey Rublev also had no trouble in his second round match. The seventh seed cruised through his clash with Thiago Monteiro in straight sets, 6-4, 6-4, 7-6 (10).
Russia’s third best hope at this year’s Australian Open – Karen Khachanov – took less than two hours to take care of his match with Ricardas Berankis.
Khachanov, who fell to Nick Kyrgios in five sets in last year’s third round, will have to get past Berrettini if he’s to achieve a new career best at the first grand slam of the year.
Talented Spanish teen Carlos Alcaraz couldn’t make it two from two in his grand slam debut, falling to Sweden’s Mukael Ymer in four sets.
“It’s a tough day for me,” the 17-year-old tipped to succeed Rafael Nadal said.
“I don’t want to lose ever. Yeah, I learn out of this match, I didn’t manage the nerves.
“I lost control of my serve, and I think for the next tournament I will try to be focused all the time, don’t complain, and yeah, I think I have to do better when things are not going in a good way.”
Sam is a sports reporter for The Sydney Morning Herald.