Fourteenth seed Radwanska crashed out 6-2 3-6 6-1 to Beck, who is currently ranked just 83 in the world.
Overcast conditions greeted home favourite Alize Cornet for the start of the action and despite a sparse lunchtime crowd she treated the home fans who had taken their seats to a comeback win against Roberta Vinci.
The French number one, saddled with the pressure of trying to provide a home winner for the first time since Mary Pierce in 2000, looked on course for a fifth consecutive defeat by Vinci before recovering to win 4-6 6-4 6-1.
She was relieved to avoid the same fate as her Fed Cup team mate Caroline Garcia who lost in the first round on Sunday.
Garcia had seemed nervous about playing on Chatrier court but not so Cornet.
"I was really glad that this court was chosen for me today," the 29th seed. "The ambience on this court is amazing.
"I think that's why I could actually come back in this match, because the people were behind me. It's just a great atmosphere in this court."
Women's eighth seed Carla Suarez Navarro underlined her "dark horse" tag with an impressive 6-2 6-2 win over Monica Niculescu of Romania.
American veteran Venus Williams, seeded 15, lost, although a 7-6 (5) 6-1 defeat by compatriot Sloane Stephens barely registered on the shock scale.
Results from the French Open Women's Singles Round 1 matches on Monday
Denisa Allertova (Czech Republic) beat Johanna Konta (Britain) 7-6(17) 4-6 6-2
Sloane Stephens (US) beat 15-Venus Williams (US) 7-6(5) 6-1
12-Karolina Pliskova (Czech Republic) beat Zhang Shuai (China) 7-6(4) 6-4
Tsvetana Pironkova (Bulgaria) beat 22-Barbora Strycova (Czech Republic) 7-6(5) 6-2
Andreea Mitu (Romania) beat Alize Lim (France) 6-3 6-2
Carina Witthoeft (Germany) beat Katerina Siniakova (Czech Republic) 6-3 7-5
Heather Watson (Britain) beat Mathilde Johansson (France) 6-4 7-5
17-Sara Errani (Italy) beat Alison Riske (U.S.) 7-6(1) 2-6 6-0
Lucie Hradecka (Czech Republic) beat Zheng Saisai (China) 6-3 6-0
30-Irina Begu (Romania) beat Bethanie Mattek-Sands (U.S.) 6-4 7-6(6)
2-Maria Sharapova (Russia) beat Kaia Kanepi (Estonia) 6-2 6-4
27-Victoria Azarenka (Belarus) beat Maria Teresa Torro (Spain) 6-2 6-1
Yulia Putintseva (Kazakhstan) beat Aleksandra Krunic (Serbia) 3-6 6-3 6-3
Ana Konjuh (Croatia) beat Margarita Gasparyan (Russia) 6-1 6-4
Daria Gavrilova (Australia) beat Johanna Larsson (Sweden) 6-1 7-6(3)
Vitalia Diatchenko (Russia) beat Stefanie Voegele (Switzerland) 4-6 6-3 6-2
Annika Beck (Germany) beat 14-Agnieszka Radwanska (Poland) 6-2 3-6 6-1
Virginie Razzano (France) beat Veronica Cepede Royg (Paraguay) 2-6 6-4 6-2
26-Samantha Stosur (Australia) beat Madison Brengle (U.S.) 6-1 6-3
19-Elina Svitolina (Ukraine) beat Yanina Wickmayer (Belgium) 6-2 6-2
11-Angelique Kerber (Germany) beat Timea Babos (Hungary) 6-0 6-1
Paula Kania (Poland) beat Mona Barthel (Germany) 5-7 6-2 6-4
29-Alize Cornet (France) beat Roberta Vinci (Italy) 4-6 6-4 6-1
Amandine Hesse (France) beat Jarmila Gajdosova (Australia) 2-6 6-3 6-2
Alexandra Dulgheru (Romania) beat Nicole Gibbs (U.S.) 6-3 6-1
8-Carla Suarez Navarro (Spain) beat Monica Niculescu (Romania) 6-2 6-2
Ajla Tomljanovic (Australia) beat Casey Dellacqua (Australia) 6-2 6-2
20-Sabine Lisicki (Germany) beat Monica Puig (Puerto Rico) 6-3 6-2