Joe Carr Sr. was the University of Kentucky's first All-American wrestler in 1975, while being coached by his brother, Fletcher, and a three-time Southeast Conference champion. He finished third at the NCAA Division I Wrestling Championships in 1975 and 1976.

He was a Junior World Team silver medalist, a two-time USA national champion, a USA World Team alternate, and a Pan American Games silver medalist.

Carr coached wrestling at Woodford County High School for 31 years. He coached 43 individual state champions, 129 state place-winners and seven All-Americans. Teams he led or assisted won eight Kentucky state team titles and finished in the Top 10 23 times. The Kentucky Wrestling Association named him Coach of the Year three times.