After dropping the first set, the No.3-seeded Radwanska faced some tricky moments early on in the next two sets - she faced break points in her first two service games of the second set, and actually lost serve in the first game of the third. But she stayed cool, calm and collected through it all and got her nose in front again every time, eventually making it past the Swiss in two hours and 14 minutes.
"The first match is always tricky, especially because each week we have different conditions, different courts and different balls," Radwanska said afterwards. "I needed this match just to feel the rhythm.
"The courts here are much slower than in Tokyo. I just had one practice so far here, and I don't think I really adjusted yet. But I think I was definitely playing better in the third set, and was winning the points in the important moments. I managed to come back from 0-40 in some of those games, as well.
"It could have gone either way - I'm just happy I could come back and win the last couple of points."
Radwanska was playing Voegele for the first time at this level, but they did have a history together.
"I never played her but I've known her for many years. We played all the junior tournaments together, under 12, 14, 16 - so we did know each other for many years, but we just never played each other."
The fans were very supportive of Radwanska all match - she was the champion here in 2011, after all.
"It's always great to see a lot of fans here, even on the practice courts, and we're always playing in a crowded stadium," she said. "I really enjoy it here and I always look forward to my matches here.
"But it's never easy here. It's always a very good tournament. Everybody is playing here. Pretty much from the first round we have to face good players. I know I always have to play my best tennis from the start. I just go and try to play my best tennis every tournament though. I hope I can win here again."
Next up for Radwanska is Madison Keys, who put on a serving display - eight aces, zero double faults - and took out Dominika Cibulkova, 64 63. Cibulkova was just a few spots out of this week's seedings.
Radwanska has beaten Keys in both of their previous meetings, though their last meeting was anything but straightforward - Radwanska barely beat the American at Wimbledon this year (read more here).
Other early winners were Lucie Safarova, who pulled off a 75 63 upset over No.15 seed Samantha Stosur, her eighth win in 10 career meetings against the 2011 US Open champion, and Kaia Kanepi, who rallied from 5-3 down in the first set and 4-2 down in the second set to beat Julia Goerges, 75 64.