Germany's Beck lost the first set but rallied to secure a 3-6 6-2 6-2 win over the 2014 French Open semi-finalist Bouchard.
Spanish qualifier Sara Sorribes Tormo sailed past Ekaterina Makarova to secure a shock second-round spot at Indian Wells.
Tokyo 2020
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Makarova, a semi-finalist at the Australian Open as recently as 2015, saw the 20-year-old Tormo secure a 6-4 6-2 victory.
Japan's Naomi Osaka overcame compatriot Risa Ozaki 6-4 6-2 while Katerina Siniakova survived a scare to beat world number 98 Mona Barthel of Germany.
Czech Republic's Lucie Safarova was a comfortable 6-3 6-2 winner against Spain's Lara Arruabarrena and Jelena Jankovic beat Irina Falconi 6-2 3-6 6-3.
Shelby Rogers emerged as a 6-4 7-6 (7/4) winner in the all-American clash against Jennifer Brady while their compatriot Varvara Lepchenko beat Yaroslava Shvedova 4-6 6-2 6-4.
Another American, Lauren Davis, took almost three hours to triumph 6-4 4-6 7-5 against Viktorija Golubic in California.
In other first-round matches, Mariana Duque-Marino beat Patricia Maria 6-4 0-6 6-4, Magda Linette saw off Taylor Townsend 6-4 6-2 and Julia Goerges triumphed 7-5 6-2 against Ajla Tomljanovic.
There was a surprise win for Shuai Peng over Lesia Tsurenko while Donna Vekic also upset the odds in beating Alison Riske.
Pauline Parmentier made a safe passage into round two while Andrea Petkovic benefited from Vania King's retirement to set up a second-round clash with second seed Angelique Kerber.
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