Stony Brook, New York, U.S.
In 1997, while still in high school, Drew voiced Stacy Rowe on the animated series Daria. She also voiced that character in the Daria television films Is It Fall Yet? and Is It College Yet?. In 2001, she made her professional stage debut as Juliet in Romeo and Juliet at the McCarter Theatre in Princeton, New Jersey. She made her Broadway debut in 2003 in Vincent in Brixton, which later took her to London's West End. She made the move to television with a guest role in the series Wonderfalls, and was in the film Radio. She appeared as Katie Burrell, the daughter of a Japanese relocation camp sergeant, in the 2007 film American Pastime.
From 2004 to 2006, Drew starred as Hannah Rogers in The WB drama series Everwood. She later guest-starred on Cold Case, Law & Order: Special Victims Unit, Medium, Castle, Glee, Supernatural, and Private Practice. She starred in the Hallmark Hall of Fame film Front of the Class (2008), and from 2008 to 2009, she had a recurring role as Kitty Romano in the AMC drama series Mad Men. In 2009, Drew was cast as Dr. April Kepner in the sixth season of the medical drama series Grey's Anatomy. In 2014, she starred in the film Moms' Night Out.