I looked this up several years ago when someone else asked, and here's what I found (but I'm not positive all of this is correct or properly up to date):
It's a five-way tie. Well, actually four guys, since one of them did it twice. In each case, the winner won three five-setters:
Gustavo Kuerten: 1997 Roland Garros
Final - Sergi Bruguera - 6-3, 6-4, 6-2
SF - Filip Dewulf - 6-1, 3-6, 6-1, 7-6(4)
QF - Yevgeny Kafelnikov - 6-2, 5-7, 2-6, 6-0, 6-4
R16 - Andrei Medvedev - 5-7, 6-1, 6-2, 1-6, 7-5
R32 - Thomas Muster - 6-7(3), 6-1, 6-3, 3-6, 6-4
R64 - Jonas Bjorkman - 6-4, 6-2, 4-6, 7-5
R128 - Slava Dosedel - 6-0, 7-5, 6-1
Stefan Edberg: 1992 US Open
Final - Pete Sampras - 3-6, 6-4, 7-6(5), 6-2
SF - Michael Chang - 6-7(3), 7-5, 7-6(3), 5-7, 6-4
QF - Ivan Lendl - 6-3, 6-3, 3-6, 5-7, 7-6(3)
R16 - Richard Krajicek - 6-4, 6-7(6), 6-3, 3-6, 6-4
R32 - Jonas Svensson - 6-4, 6-2, 6-2
R64 - Jakob Hlasek - 7-5, 6-2, 6-1
R128 - Luiz Mattar - 6-4, 7-5, 6-2
Gaston Gaudio: 2004 Roland Garros
Final - Guillermo Coria - 0-6, 3-6, 6-4, 6-1, 8-6
SF - David Nalbandian - 6-3, 7-6(5), 6-0
QF - Lleyton Hewitt - 6-3, 6-2, 6-2
R16 - Igor Andreev - 6-4, 7-5, 6-3
R32 - Thomas Enqvist - 6-0, 6-4, 6-7(5), 6-4
R64 - Jiri Novak - 2-6, 6-4, 6-4, 5-7, 6-3
R128 - Guillermo Canas - 6-2, 2-6, 4-6, 6-3, 6-2
Bjorn Borg: 1974 Roland Garros
Final - Manuel Orantes - 2-6, 6-7, 6-0, 6-1, 6-1
SF - Harold Solomon - 6-4, 2-6, 6-2, 6-1
QF - Raul Ramirez - 6-2, 5-7, 4-6, 6-2, 6-3
R16 - Erik Van Dillen - 0-6, 6-3, 6-3, 5-7, 6-3
R32 - Jean-Loup Rouyer - 6-4, 6-2, 6-0
R64 - Toma Ovici - 6-1, 6-1
R128 - Jean-Francois Caujolle - 4-6, 6-0, 6-4
Bjorn Borg: 1977 Wimbledon
Final - Jimmy Connors - 3-6, 6-2, 6-1, 5-7, 6-4
SF - Vitas Gerulaitis - 6-4, 3-6, 6-3, 3-6, 8-6
QF - Ilie Nastase - 6-0, 8-6, 6-3
R16 - Wojtek Fibak - 7-5, 6-4, 6-2
R32 - Nikola Pilic - 9-7, 7-5, 6-3
R64 - Mark Edmondson - 3-6, 7-9, 6-2, 6-4, 6-1
R128 - Antonio Zugarelli - 6-4, 6-2, 9-7