Cox transferred to the University of Arkansas from Tulsa University following the 1977 season. After sitting out 1978 due to NCAA transfer rules, Cox added the punting job to his kicking duties. Steve was named All-Southwest Conference in 1979 and 1980, and was also named an All-American in 1980 by the UPI, the Sporting News, College & Pro Football Newsweekly, and Football News after leading the nation in punting by averaging 46.5 yards per punt.
Cox was inducted into the Arkansas Sports Hall of Fame in 2004.
Though primarily a punter and kickoff specialist, Cox was also used for long field goal attempts. On October 21, 1984, Cox kicked a 60-yard field goal in a game against Cincinnati. The kick was the second-longest field goal at the time (behind Tom Dempsey's record kick), but has since been matched by Morten Andersen, Rob Bironas, and Dan Carpenter and surpassed by Justin Tucker, Jason Elam, Matt Bryant, David Akers and twice by Sebastian Janikowski. Cox's kick remains one of only 26 field goals of 60 yards or more in NFL history and one of only two which was done with the straight-ahead style (the other being Tom Dempsey).