Beck ousted fifth-seeded Lucie Safarova 6-2, 4-6, 6-4. She dropped her serve at least once in each set but had six breaks herself, including three in the first set and two in the third.
Beck took nearly two-thirds of the points in the opening set and held on in the third set by finishing off two of her three break chances.
Andrea Petkovic outlasted seventh-seeded Eugenia Bouchard 2-6, 7-5, 6-1 while qualifier Tereza Smitkova defeated Kristina Mladenovic 7-5, 6-1 in the tournament's final first-round matches.
The formal name of the tournament is the BGL BNP Paribas Luxembourg Open.