He has the third most career points for a Brown behind fellow kickers Phil Dawson (second) and Lou Groza.

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.

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.

Later NFL players to have this dual role were Steve Little of the St. Louis Cardinals in 1979, Russell Erxleben of the New Orleans Saints briefly in 1979 and 1980, and Frank Corral for the Los Angeles Rams in 1980 and 1981.