Crow was born on November 10, 1986, in Topeka, Kansas to parents Kevin and Julie Crow. Crow and his siblings—brother Travis and sister Jennifer—were raised in the small community of Wakarusa, Kansas, not far from Topeka. Following his graduation from Washburn Rural High School, Crow attended the University of Missouri.
In his three years for the Missouri Tigers baseball team, Crow started 46 games, going 23-8 with a 3.27 earned run average (ERA).
Spending time in both the bullpen and the starting rotation as a freshman, Crow earned his first career victory by throwing a complete game against Pepperdine, staving off elimination in the 2006 NCAA Regional. Mizzou went on to win the regional, becoming the first #4 seed ever to win an NCAA Regional.
As a sophomore, Crow went 9-4 with a 3.60 ERA, earning first-team All Big 12 honors.
As a junior, Crow went 13-0 with a 2.35 ERA. He threw four complete-game shutouts and struck out 10.65 batters per nine innings. He was named the Big 12 Conference Pitcher of the Year.
He won the Robert A. McNeece Award as the top professional prospect in the 2007 Cape Cod Baseball League season while playing for the Falmouth Commodores.
Crow was selected by the Washington Nationals in the first round of the 2008 Major League Baseball Draft with the ninth overall selection. Negotiations stalled and Crow did not sign. Crow signed with the Fort Worth Cats for the 2009 season.
Crow was selected with the twelfth pick in the first round of the 2009 Major League Baseball Draft by the Kansas City Royals. Crow signed a contract with the Royals on September 15, 2009. To make room for Danny Duffy on the Double A Northwest Arkansas Naturals roster, Crow was demoted to High-A Wilmington on July 31, 2010.
Crow made his first major league appearance on March 31, 2011, which was Opening Day. He faced four Angels batters, striking out three.
On May 30, 2011, Royals manager Ned Yost announced that Crow had been promoted to the team's closer position on a temporary basis to replace the struggling Joakim Soria. However, on June 6, Yost announced that Soria had earned the spot back. Crow had no save opportunities in his brief stint as closer.
In 2011, Crow was selected to the All-Star Game, although he did not play.
On November 28, 2014, the Royals traded Crow to the Miami Marlins for Brian Flynn and Reid Redman.
Crow signed a minor league deal with the Chicago Cubs in February 2016. He became a free agent on November 7, 2016.
Crow is a sinkerballer with a heavy sinker at 94–97 mph. His main off-speed pitch, and most-used pitch against right-handed hitters, is a slider at 85–88. He also has a four-seam fastball. Against left-handed hitters, he throws a small amount of curveballs and changeups. The majority of his 2-strike pitches are sliders, owing to its 49% whiff rate.