Celestini in 2010
|Date of birth||31 October 1975|
|Place of birth||Lausanne, Switzerland|
|Height||1.83 m (6 ft 0 in)|
|* Senior club appearances and goals counted for the domestic league only.
Having started and finished his 15-year professional career with Lausanne, he also played one full decade combined in France and Spain, representing four teams.
Born in Lausanne, Celestini started playing professionally with hometown's FC Lausanne-Sport, then had a four-season spell in France, spending two years each with Troyes AC and Olympique de Marseille. Whilst with the latter he played the 2004 UEFA Cup final, coming on as a substitute in the second half of the 0–2 loss to Valencia CF.
Celestini moved to Spain in 2004, playing one season with Levante UD. After the team's immediate relegation he stayed in the country, joining fellow La Liga club Getafe CF where he never was an undisputed starter, but managed to feature heavily in consecutive seasons. In 2007–08, as the Madrid outskirts side reached the quarter-finals in the UEFA Cup, he scored against R.S.C. Anderlecht in a 2–1 group stage home win.
On 17 February 2010, after having been regularly used in the past two seasons, albeit not as a usual first-choice, 34-year-old Celestini announced he would not renew his contract with Getafe, choosing to return to Lausanne on a one-year deal. However, due to a break in negotiations over his future role at the club after his playing career, he decided to retire before the end of the campaign, playing his last game on 15 December against U.S. Città di Palermo for the Europa League group stage.
After a spell as assistant manager at Spanish club Málaga in 2013–14 and manager at Italian Serie D side Terracina, on 24 March 2015 Fabio Celestini replaced Marco Simone as manager of Swiss club FC Lausanne-Sport, then in a poor performance run in the Challenge League, the second tier of Swiss football. Two months later, he was confirmed in this role for the next three years. The following season, Celestini successfully led Lausanne-Sport to win the 2015–16 Challenge League, which promoted them back to the Super League, the top tier of Swiss football, for the 2016–17 season. 
- Marseille move cheers Celestini; UEFA.com, 30 July 2002
- Levante capture Celestini; UEFA.com, 18 August 2004
- Getafe take top spot with Anderlecht win; UEFA.com, 19 December 2007
- Degen krank – Celestini zurück (Degen ill – Celestini returns); SF Sport, 16 February 2010 (German)
- "Le FC Lausanne-Sport limoge Simone et mise sur Celestini" (in French). 24 March 2015. Retrieved 22 December 2016.
- "Celestini prolonge trois ans au FC Lausanne-Sport" (in French). 21 May 2015. Retrieved 22 December 2016.
- "Lausanne accède à l'élite" (in French). 5 May 2016. Retrieved 28 November 2016.