A native of the United States and the son of Canadian Reggie Cleveland, who pitched in Major League Baseball for 13 years, Jon Cleveland began swimming at the age of nine but eventually went north of the border to compete for Canada on the international stage. His first major tournament was the 1988 Summer Olympics, where he finished seventh in the 200 metres breaststroke. He captured bronze in the event at the 1989 Pan Pacific Championships, as well as silver in the 4x100 m medley relay, and placed fourth in the 100 metres breaststroke. He was fourth again in the 100 m at the 1990 Commonwealth Games, but took gold in the 200 m and the 4x100 m relay alongside Marcel Gery, Tom Ponting, and Mark Tewksbury. The quartet earned silver in this event at the 1991 Pan Pacific Championships (with the assistance of Stephen Clarke), but finished fourth in that year’s World Championships. Cleveland, meanwhile, was fifth in both the 100 and 200 m at the former tournament, and eleventh in the 50 m breaststroke and eighteenth in the 200 m at the latter. His individual fortunes did not improve at the 1992 Summer Olympics, where he finished 13th and 14th in the 100 and 200 m respectively, but his 4x100 m relay squad (again with Clarke) won the bronze medal. Along with Ponting, Tewksbury, and Stephen Vandermeulen, he also set a world record in the 4x100 m relay that year.
Cleveland was less successful in 1993, placing sixth and eighth in the 4x100 m relay and the 200 m respectively at the World Short Course Championships, and second, fourth, and fourth in the 200 m, 100 m, and 4x100 m relay at the Pan Pacific Championships. In 1994 he was 10th, 11th, and 12th in the 4x100 m relay, the 100 m, and the 200 m at the World Championships, but captured three medals at Commonwealth Games: silver in the 4x100 m relay (alongside Clarke, Rob Braknis, and Chris Renaud) and bronze in the 100 and 200 m breaststroke. At the 1995 Pan American Games he captured silver in the 100 m, bronze in the 4x100 m relay, and finished fifth in the 200 m. In his final major international tournament, the 1996 Summer Olympics, he was 23rd in the 100 m, 15th in the 200 m, and 12th in the 4x100 m relay (with Braknis, Clarke, and Eddie Parenti). He retired after the Games and took up news production and anchoring and coaching.