He played for HC Vitkovice from 1977 to 1988 and was their captain for part of that time. Svozil then went to Germany and Italy where he played until his retirement in 1995. He played for Czechoslovakia in the 1977 World Junior Championship and the 1979 and 1983 World Championships. He was the coach of HK SKP Poprad from 2004 to 2006 and then was rehired in 2011. He was drafted in the 12th round, 194th overall in the 1978 NHL Amateur Draft by the Detroit Red Wings.