Alexander Zverev will also be in the mix but the German fourth seed had a close shave on Saturday, needing three sets to advance past Russian Daniil Medvedev 6-4 1-6 7-6(5), while Denis Shapovalov also moved on by defeating Bosnian Damir Dzumhur.
Kokkinakis had trained with Federer but had never played the Swiss previously.
With his vicious forehand dialled in, the Australian was able to wear down Federer, who announced after the match that he was again skipping the European clay court season.
A disappointed Federer, who loses his number one ranking to Rafa Nadal with the loss, said he just could not raise his game.
"Sometimes you have these matches,” he said. “Sometimes you find a way through. I just couldn't get it done today.”
His mood contrasted sharply with an ecstatic Kokkinakis.
“When I feel like I am playing on my terms, I don't think there's too many people who can go with me,” the 21-year-old told reporters.
Fernando Verdasco will be Kokkinakis's next opponent after the Spaniard rallied from a set down to beat countryman Guillermo Garcia-Lopez 4-6 6-0 6-2.
Zverev needed more than two hours to prevail over Russian Medvedev, taking his ninth consecutive deciding-set tie-break victory, and will next face David Ferrer.
Canadian 18-year-old Shapovalov breezed through the first set in 24 minutes against Dzumhur and led by a break twice in the second before breaking for the fifth time at 5-5 and serving out the 6-1 7-5 victory.
He will next face Borna Coric of Croatia, who turned back Argentine Leonardo Mayer 6-7(5) 6-3 6-4 in a nearly three-hour match.
Japanese left-hander Yoshihito Nishioka, coming back after a year out due to major knee injury, went down 6-1 6-4 to 10th seed Tomas Berdych.
Britain's Kyle Edmund also lost, falling 7-6(4) 4-6 7-6(5) to American Frances Tiafoe, who frittered away four match points in the tiebreak before finally winning.
On other matches, Kevin Anderson and Jack Sock advanced with straight set wins. Australian Nick Kyrgios took down Dusan Lajovic 6-4 6-1, while Pablo Carreno Busta defeated Denis Istomin 6-1 6-0.
(Reporting by Gene Cherry in Salvo, North Carolina; Additional reporting by Jahmal Corner; Editing by Peter Rutherford)
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