He reached the final of the World Amateur Snooker Championship in 1978, losing to Cliff Wilson. After becoming a professional in 1979, his most notable early achievement was to reach the final of the untelevised 1983 Professional Players Tournament, which he lost 9-8 to Tony Knowles.
In 1986, Johnson arrived at the World Championship having never won a match at the Crucible Theatre and as a 150-1 outsider. He beat Terry Griffiths 13-12 in the quarter-finals after trailing 12-9, and then Knowles in the semi-finals. In the final he met world number 1 Steve Davis - then at the peak of his ability - and defied all odds to win 18-12, although Davis scored more points in total A passionate fan of Bradford City football club, he wore a t-shirt with the slogan "Bradford's Bouncing Back" (a reference to the Bradford fire a year earlier) whenever he was not playing during the tournament.
Johnson's best showing during his season as world champion was a solitary semi-final, but he defied expectations at the 1987 World Championship and reached his second final, again meeting Davis but this time losing 18-14. He reached number 5 in the world rankings in the 1987-88 season, largely as a result of his success at the Crucible.
Johnson won the Langs Scottish Masters in 1987, beating Terry Griffiths 9-7 in the final and taking his only other major snooker title. He rarely came close to repeating his successes, and rapidly descended the rankings, dropping out of the top 16 by 1991 and the top 32 by the mid-1990s. His last appearance at the World Championship came in 1991. He suffered heart and eye problems during the 1990s, and retired from professional snooker at the end of the 2003/2004 season.
Johnson remains the player who came closest to beating the "Crucible Curse", in which no first-time world champion has ever successfully defended the title. Johnson's defence saw him both reach the final and come within 4 frames of victory. Ken Doherty also reached the final in 1998,a year after his first win at the crucible, but lost by a greater margin to John Higgins. No other first-time champion has reached the final.
Johnson also won the Seniors Pot Black Trophy in 1997, beating Terry Griffiths in the final. He was also an early influence and friend of the snooker player Paul Hunter. He is now a commentator for Eurosport.