Hayen, 46, lasted just four months in charge of Burnley and was sacked with the club sitting down in the Championship relegation zone.
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Burnley have now confirmed the appointment of former Basel and CSKA Moscow boss Celestini, 50, for an as yet undisclosed number of years.
Speaking to the club’s website, Celestini said: “I look at Burnley and I see a club with big potential and players with real quality, who can help us achieve our supporters’ ambitions.
“I know the situation we are in, and I have lived it before. Our supporters deserve to see a team playing with belief from the first whistle.”
Celestini will take charge of his first game on Saturday (October 31) as Burnley welcome Portsmouth to Turf Moor.