Peer had given Radwanska trouble in the past, winning three of their six previous meetings - but there was nothing troublesome about this one, as the No.4-seeded Radwanska rolled to a 61 61 victory, maintaining some impressive consistency throughout the match with 13 winners to just eight errors.
It was even more impressive in that she lost early in both of her lead-up events, Madrid and Rome, and pulled out of Brussels last week with a right shoulder injury. It all seems just fine now, though.
"My shoulder was worse after Rome, so I decided to have a week of rest last week, and just practice and do rehab before Roland Garros," Radwanska explained. "I was practicing really hard last week to be ready for the French Open, which is of course the most important tournament on clay for us.
"I'm definitely ready to go now and am 100% fit. I think it was a good move."
The location of that most important clay court tournament doesn't hurt. "Definitely Paris is one of my favorite cities," Radwanska said. "I think the atmosphere here is very cool. You can just go out and walk around the city. You can just relax a little bit and forget about tennis when you're here."
Her favorite place to go in the French capital? "Champs-Élysées of course," she said.
Radwanska didn't just turn heads with her sharp shotmaking at Roland Garros on Monday, she was also sporting a new look - she recently changed her hair color to blonde. "Well, it's just to change something," she said. "My sister went to brunette for I think two weeks, but she didn't like it at all, so she changed back to blonde again. So we're both blonde now. But I think I'll go back to dark hair because everybody is telling me to come back to the dark, so I think I will do it soon, probably."
Awaiting Radwanska in the second round will be Mallory Burdette, who won her first round match on Sunday. The World No.4 was asked about the American rising star in her post-match presser.
"To be honest, I don't know much - nothing at all, actually," Radwanska said. "I might Google her. I don't know. Actually I didn't even see the draw, so I don't even know who she beat in the first round. Maybe I will look at that. But it's always a good challenge to play someone you don't know."
Burdette knew all about Radwanska, though. "She's a fantastic player," she said. "She's had a great past couple years. I know she's extremely fit and very fast. She's super solid, so it should be an interesting match-up. I'll definitely watch a little bit more of her over the next couple of days before we play and just go from there, but I know she's a great competitor and it will be a good match."