After graduating from the Emory University, Sarbh worked with the Alliance Theatre in Atlanta for a year as a literary intern. He performed around Atlanta in the 2009 productions of such plays as The Show!, The Breakup, Tennis in Nablus, and Ice Glen; the last of the roles won him a Major Supporting Role Metropolitan Atlanta Theatre Award. Sarbh moved back to Mumbai in 2012 and began acting in local theatre productions. His performance as Happy Loman in Alyque Padamsee’s 2013 revival of Death of a Salesman drew the attention of several commentators. Sarbh continued to act in Mumbai-based plays including Rajat Kapoor’s What’s Done is Done, Rage Productions’s The Glass Menagerie, Vickram Kapadia’s The Merchant of Venice, Kalki Koechlin’s Living Room, and later made his directorial and writing debut with the 2014 production Bull and Eat respectively. He was listed in Forbes India’s 30 under 30 list in 2015 for his contributions to the Mumbai theatre industry.

Sarbh had two Bollywood releases in 2017–the thriller A Death in the Gunj and the romance film Raabta. He also starred in Natasha Mendonca’s directorial debut Ajeeb Ashiq, an independent film that focuses on the gender identity and human sexuality. The production was screened at various film festivals across Europe. He played the role of Malik Kafur in Sanjay Leela Bhansali’s 2018 epic romance Padmaavat, featuring alongside Shahid Kapoor, Deepika Padukone and Ranveer Singh. In 2019, he had a role in the Netflix film House Arrest, and the following year featured in Yeh Ballet, another Netflix film.

In 2020 he appeared in Bejoy Nambiar’s Taish releasing on ZEE5 in October.