The Irish rugby club announced on Sunday that 42-year-old Foley had passed away in the team's hotel prior to their European Champions Cup clash against Racing 92.
As a result of the tragic news, the match, which was due to go ahead in Paris on Sunday afternoon, was postponed.
On Monday, Foley's relatives released a statement, which read: "The family of Anthony Foley wishes to extend its deepest appreciation to the endless legions of friends and the wider rugby family here and abroad for the huge outpouring of support and sympathy since Anthony's tragic and most untimely passing at the weekend.
"With Anthony's passing, we have ultimately lost an amazing, adoring and loving father and husband; an equally caring, loyal and devoted son and brother; a central and go-to figure for the wider Foley and Hogan families. Our anguish at the sudden loss of Anthony is bottomless. We have been plunged deep into an incomprehensible darkness and sense of loss that we must work our way through over the coming days, weeks, months and years.
"We know, too, that his sudden death has brought the rugby worlds of Shannon RFC, Munster, Ireland and much further afield crashing down. You have lost a former player, coach, friend and all-round inspiration - your and our hero both. We mourn his loss together.
"We again wish to thank everyone for their support; it will help carry us through these darkest days."
Foley, who won 62 caps for Ireland during his playing career, took over as Munster head coach in 2014.