In his first professional match, at the 1978 UK Championship, he lost 8–9 to Rex Williams after leading 8–1. However, he could hardly have expected what would come in the 1979 World Championships. After qualifying he beat Perrie Mans and Alex Higgins. Interviewed after beating Eddie Charlton in a long semi-final, it suddenly dawned on him what he had done, and he said "I'm in the final now, you know!" in his broad Welsh accent. He went on to beat Dennis Taylor 24–16 in the final, becoming world champion at the first attempt. In the same year he was part of the Welsh team that won the inaugural World Cup of snooker: he, Ray Reardon and Doug Mountjoy beat England 14–3 in the final. But at the end of 1979, he lost 13–14 in the UK Championship final to John Virgo.
1980 started well for Griffiths as he won the Masters, beating Alex Higgins 9–5 in front of 2,323 spectators at the Wembley Conference Centre. It was his first appearance at the Masters and turned out to be his only win there. He then won the Irish Masters also at the first attempt, beating Doug Mountjoy 9–8, but the Crucible Curse struck at the World Championships that year, as he lost his second-round match (which was then a 'first round' for the top eight players, who had byes) to the up-and-coming Steve Davis.
He retained the World Cup later on in 1980 for Wales and again won the Irish Masters in 1981 before losing to eventual winner Steve Davis again in the World Championship.
He also lost 3–16 to Davis in the UK Championship final in 1981, beginning a six-month period in which he and Davis faced each other in almost every major tournament final. Griffiths triumphed twice, in the Classic in early 1982 and later the Irish Masters (becoming the first player to win three consecutive titles) beating Davis on both occasions (the Classic 9–8 and the Irish Masters 9–5). Unsurprisingly, after Davis was sensationally beaten by Tony Knowles in the first round of the 1982 World Championship that year, Griffiths was immediately installed as the bookmakers' favourite for the title. However, a second surprise followed as Griffiths was beaten, also in the first round, by Willie Thorne. He never again won a ranking event, although he won several major invitational events; the 1984 Malaysian Masters, where he topped a round robin group (Tony Meo was the runner up), the 1984 Singapore Masters, where also topped a round robin group (Davis was runner up), the 1985 Hong Kong Masters, where he beat Davis 4–2; the 1986 Belgian Classic, where he beat Kirk Stevens 9–7 in the final, in an event featuring 8 of the top 9 in the year's rankings.
At the end of 1982, he won the UK Championship, beating Alex Higgins in a classic 16–15 final. He did take the Pot Black title in 1984 and the Welsh Professional Championship between 1985, 1986 and 1988. He reached the final of the World Snooker Championship again also in 1988, but he lost to old rival Steve Davis 11–18. During the final session of the championship, he accidentally knocked over a globe on the Crucible set, denting part of it with his foot: he was awarded the globe at the end of the match. He achieved the notable feat of reaching at least the quarter-finals of the World Championships for nine consecutive years between 1984–1992.
By the 1990s he began to struggle in the rankings but he still reached the semi-final of the 1992 World Championship, with victories over Bob Chaperon, Neal Foulds and the young Peter Ebdon, losing to Stephen Hendry. Having lost at the Crucible in 1996 against his old rival Steve Davis (who he never beat at the Crucible in 7 attempts) in the last 16 (after beating the young Scottish player Jamie Burnett in a final frame decider 10–9 in the first round, having trailed 0–6 and 5–9), he immediately announced his retirement from the game. However, he did play at the Crucible one last time in 1997 losing to fellow countryman and debutant Mark Williams, in another final-frame decider, 9–10. This meant that he played a total of 999 frames at the Crucible
White's third ranking win – the 1987 British Open – helped him to end the 1986/1987 season as World number 2, behind Steve Davis who defeated him 16–11 in the semi-finals of the 1987 World Championship. Later that year White and Davis contested a memorable UK Championship final which Davis won 16–14.
In 1988 he defeated John Campbell, Stephen Hendry and Tony Knowles to reach his fourth World Championship semi-final. He played Terry Griffiths and, trailing 11–13, lost a tied frame on a respotted black. Griffiths went on to reach the final courtesy of a 16–11 win. White did at least manage to consolidate his number-2 world ranking. However the 1988/1989 season was less successful, and White's ranking slipped. He trailed John Virgo 11–12 in the second-round of the 1989 World Championship and looked beaten when his opponent was on a break of 26 in the following frame. Virgo, however, called a foul on himself and White was able to win 13–12. The reprieve was short-lived as White was soundly beaten 7–13 by eventual finalist John Parrott in the quarter-finals. White avenged this defeat later in the year by beating Parrott 18–9 in the final of the invitational World Matchplay.
In 1990 White recorded an 16–14 victory over Steve Davis in the semi-finals of the World Championship. It was Davis's first defeat in the event in 4 years. White subsequently lost his second World Championship final 12–18 to Stephen Hendry. However White beat Hendry 18–9 to retain his World Matchplay title later in the year and this win was followed by a 10–4 victory over Hendry (after leading 9–0) in the final of the 1991 Mercantile Classic. White continued his run of success with victory in the short-lived World Masters, beating Tony Drago 10–6 in the final.
Steve James ended Hendry's reign as World Champion in the 1991 World Championship and White in turn defeated James to reach the final. He played John Parrott and was whitewashed in the first session 0–7. Although White managed to close the gap to 7–11, Parrott was able to seal a convincing 18–11 victory. Parrott then overcame White 16–13 to win the UK Championship later in 1991.