Hsu was born to a single mother in Torrance, California. Her maternal grandmother had left mainland China and sent Hsu's mother to the United States as a teenager for emotional reasons. Hsu attended Palos Verdes Peninsula High School. She moved to Brooklyn to pursue theatre and graduated from NYU Tisch School of the Arts in 2012. She also trained with the Atlantic Theater Company.
Hsu began her career in experimental theatre and comedy. From 2013 to 2015, Hsu made regular appearances on the MTV reality comedy series Girl Code. She landed her first recurring television role as Joy Armstrong in the Hulu series The Path.
Hsu was called in for the first table reading of The SpongeBob Musical in 2012 to read for Karen, the anthropomorphic computer. She would go on to play the character on stage in Chicago in 2016 before making her Broadway debut in 2017.

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