Cockroft served as the Browns' primary punter and placekicker for the first nine seasons of his career. In 1977, he dropped punting from his duties and became solely a placekicker. He and Tampa Bay Buccaneers kicker/punter Dave Green were two of the last NFL players to lead their teams in both punting and kicking in the same season (1976).
He was involved in the January 4, 1981, American Football Conference divisional play-off game versus the Oakland Raiders. Cockroft missed field goals from 47 and 30-yards in the second quarter. The Browns scored a touchdown on a 42-yard interception by Ron Bolton with 6:02 left in the second quarter, but the extra point attempt by Cockroft was blocked. Cleveland would lose the game 14–12, a game which is nicknamed Red Right 88.