Their delayed match survived the cull after being moved on to the roof-covered showcourt, opening play before defending champion Novak Djokovic and Swiss great Roger Federer took to the stage.
"Of course, now is good two matches, Novak and Roger, so I have plenty of tennis to watch," she said as she came off court, relieved to have got her match completed.
"A one-hour match is always good, especially with that weather now, we are playing every day, so I am just very happy with that win and ready for the next one."
Her path through the French Open was also blighted by rain, and lost in the fourth round before blasting organisers for forcing her to play in misty drizzle.
Radwanska's potential second-round opponents, Italy's Karin Knapp or Croatian Ana Konjuh, were still waiting for the showers to abate to get their match completed as the postponed fixtures began to pile up.
The Pole raced into an early lead, breaking twice in the four first games before being pegged back by Kozlova, who did not seem in the least bit overawed despite making her debut at Wimbledon.
Radwanska, who reached the final in 2012, has never lost a first-round match on the Wimbledon grass and she kept up that record, sealing victory on her third match point on a second serve.
She thought she had it in the bag with an ace, before a Hawkeye challenge by Kozlova left her smiling ruefully at being just millimetres short.
But having got the job done, it was now "eating or talking, watching Centre Court" as organisers hastily try to redraw the schedules.
"Just looking at the sky, if it's still raining or not. What else you can do?" she told reporters.
(Editing by Ed Osmond)
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