Drew was born in Washington, D.C. She has two brothers and her parents are both lawyers. She attended Middlebury College where she studied English Literature and Spanish. After graduating college, she moved to London and trained at the London Academy of Music and Dramatic Art. She left school a term early when she was cast in the world premiere of Neil LaBute's The Distance From Here directed by David Leveaux at the Almeida Theatre (2002).

She has appeared on television shows including Law and Order: SVU as Suzy Fleckner (2003) and CSI: Miami (2004). On CSI she played a character who was originally named Colleen Mitchell but after the producers cast Drew, they changed the character's name to Colleen Mendoza, to reflect Drew's mixed heritage. She also played Angelica Sandoval on The Inside, with Peter Coyote and Rachel Nichols, for the Fox network, but the series was never aired.

In 2005 she recurred on the HBO series Entourage and from 2006–2007 she was on the NBC show Las Vegas, playing a new member of the Montecito Casino team with James Lesure, Josh Duhamel and James Caan. While she was still working on Las Vegas, she was cast on ER as the new medical student, Katey Alvaro, who is doing her residency in the ER and has an affair with another Doctor, played by Shane West. She was on ER during seasons 11 and 12.

In 2008 Drew was voted one of Maxim's "100 Hottest Girls of Entourage" (#19). In 2009 she was voted one of ER's "Hottest Doctor's of All Time" (#2).