There was also success for Alex de Minaur who is into the quarter-final in Lyon.
The Geneva tournament's top seed, Daniil Medvedv, is out however, beaten by Richard Gasquet 6-2 7-6 (7-5). The world was making his comeback after six weeks out after a hernia operation.
Gasquet, who beat Australia's John Millman in the first round, was able to take advantage of several double faults by Medvedev who had hoped for more match practice ahead of next week's French Open.
O'Connell, who came through two rounds of qualifying, and is ranked 124 in the world, beat No.42-ranked Albert Ramos-Vinolas 7-6 (7-5) 6-4. The Australian No.9 now plays American Reilly Opelka who had a first round bye.
Kokkinakis defeated Italian veteran Fabio Fognini 6-4 6-3, ranked 33 places higher at No.52, and next meets Argentina's Federico Delbonis.
"I was down 0-40 in my first game so that was bit of a hole for me," said Kokkinakis, "but I was fortunate to make my way out of that game.
"I tried not to give him too much momentum because when he's playing good and with momentum, he's a tough man to beat.
"I'm pretty happy with it in just my second clay court tournament of the swing".
The South Australian had planned to play in Madrid after a busy playing period where he reached the fourth round in Miami, but he suffered illness and concentrated on being as fit as possible for next week's French Open instead.
His clay court season thus began last week in Rome, where he lost in qualifying.
In Lyon de Minaur recovered from a terrible start to beat France's Ugo Humbert 1-6 6-3 6-2 in two hours, 25 minutes.
The Australian No.1 saved 11 break points in the second showing extraordinary resilience - including five in an epic 16-minute fifth game with the score at 2-2.
In the quarter--final de Minaur, who is fourth-seeded, faces Yosuke Watanuke, a lucky loser but not to be underestimated. The Japanese put out No.8 seed Pedro Martinez in the first round.
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