Pudi was born in Chicago, Illinois, the son of Teresa (née Komendant), a programmer and analyst, and Abraham Pudi. His father is American of Indian origin, and his mother is Polish (born in Pokośno, Poland, to Marianna (Organek) and Boleslaw Komendant). Pudi grew up speaking Polish with his mother and grandmother. He grew up on Chicago's south side, but he attended Notre Dame College Prep in Niles, Illinois, a northern Chicago suburb. In 2013, he gave the commencement address at Notre Dame College Prep.
At Marquette University, Pudi was the first winner of the Chris Farley Scholarship. Beyond paying for a year of school, the scholarship led to him performing at an improv comedy event featuring Jim Breuer and Dave Chappelle, increasing his interest in studying improv. Pudi performed in summer stock theatre in Wisconsin after Marquette, then studied at The Second City in Chicago.
Pudi became a recruiter for an executive search firm before moving to Los Angeles in 2005 to seek television and film roles. His home-based job permitted him to attend auditions, helping him avoid the "struggling actor thing in terms of waiting tables, temping and so on". He performed in several television pilots before joining the cast of Community.
Pudi has appeared in multiple television advertisements for such products as Snickers, Verizon, McDonald's, T-Mobile, and Pokémon. In 2012, he appeared as a celebrity contestant on GSN's version of The Pyramid with fellow Community cast member Yvette Nicole Brown.