The fifth-seeded Swiss dropped the opening set on a day when play began an hour late due to rain, but quickly began gathering his resources in the second set.
With the Lausanne native levelling at a set apiece against his 18-year-old opponent, Wawrinka took control, breaking for 4-2 in the third set on his way to victory in two and a quarter hours.
The winner notched 16 aces but broke on only one of his seven chances.
Wawrinka produced 40 winners and 52 unforced errors as he moved into the third round against another Croatian, Ivo Karlovic.
French 12th seed Richard Gasquet completed this week's demolition of the Australian young guns as he backed up a win over Nick Kyrgios with victory over Kokkinakis.
Gasquet defeated the qualifier 7-6 (7/5), 6-2 to next play US Open champion Marin Cilic.
Pole Jerzy Janowicz advanced over American Jared Donaldson 6-4, 7-6 (7/4).
In the women's second round, Italian Karin Knapp beat Russian hopeful Daria Gavrilova 6-4, 6-1 while former number one Jelena Jankovic left American Madison Keys a tearful wreck during and after a 7-6 (7/4), 6-4 second-round win.
Keys was seen sobbing at mid-match during a coaching conference with Lindsay Davenport and was again caught after the loss with her face buried in a towel in a busy hallway outside the locker room.
The 19th-ranked Keys has won just two matches since achieving a Wimbledon quarterfinal.