Beck, who earned a top-50 ranking for the first time this week after her fourth quarterfinal of the season at the Hungarian Grand Prix, was broken four times but saved eight other break points.
Petra Martic of Croatia rallied to beat former champion Andrea Petkovic 6-7 (5), 7-5, 6-3. Martic will face Patricia Mayr-Achleitner in the second round. The Austrian defeated Tereza Mrdeza of Croatia 6-4, 6-3.
Viktorija Golubic of Switzerland held serve throughout to defeat No. 5 Kiki Bertens of the Netherlands 6-2, 6-2 and set up a second-round match against Andrea Hlavackova of the Czech Republic, who beat Eleni Daniilidou of Greece 6-3, 6-1.
No. 6 Chanelle Scheepers of South Africa beat Anna Schmiedlova of Slovakia 6-1, 6-4; Estrella Cabeza Candela of Spain defeated Tadeja Majeric of Slovenia 6-1, 6-3; Elina Svitolina of Ukraine beat Michaela Honcova of Slovakia 6-2, 6-2; and Alexandra Cadantu beat Maria Joao Koehler of Portugal 6-2, 6-4.