He was signed by the Ravens as an undrafted free agent in 2010. He played college football at Tennessee. Cox was a walk-on for the Tennessee Volunteers football team as a long snapper, earning a scholarship prior to the 2008 season.
Cox was the starting long snapper for Tennessee for the 2007, 2008, and 2009 seasons.
He earned Academic All-Securities and Exchange Commission honors from 2006–2009. Cox was invited to participate in the 2010 Senior Bowl that took place on January 30, 2010.
Cox was signed by the Baltimore Ravens as an undrafted free agent following the 2010 NFL Draft on May 6, 2010. He earned the starting long snapper job after the Ravens released Matt Katula on August 14, 2010.
Morgan Cox was part of the 2012 Baltimore Ravens Super Bowl XLVII victory on February 3, 2013.
On the Season opener of the 2013 season against the Denver Broncos, Cox recovered a punt that was muffed by Wes Welker. On October 19, 2014, Cox tore his Anterior Cruciate Ligament against the Atlanta Falcons and was placed on season ending injured reserve the day after. He was replaced by Kevin McDermott for the remainder of the season.
Cox recovered from the injury, and resumed his longsnapping duties for the 2015 season.
Cox, along with punter Sam Koch, was named to his first Pro Bowl in 2016. However, due to the drafting format to decide teams, Koch and Cox ended up on different teams.
On March 7, 2016, Cox signed a 5 year extension to remain with the Ravens.