Kluivert, who played for the Netherlands and clubs including Barcelona, AC Milan and Ajax Amsterdam, scored nearly 250 goals for club and country.
This week's streak began in the UEFA Youth League on Wednesday when his youngest son and 18-year-old winger Shane curled a shot home from the edge of the box to spark Barcelona's 2-1 comeback win over Paris St Germain.
A day later, 24-year-old Ruben, who is a defender, headed home the second goal in Olympique Lyonnais' 2-0 win over Red Bull Salzburg in the Europa League.
Bournemouth's 26-year-old winger Justin Kluivert completed the unique treble, netting a stunning long-range strike in Friday's 3-1 Premier League victory over Fulham.
"Proud dad," Patrick, who is now head coach of the Indonesia national team, wrote on Instagram after each of his sons' teams won their matches.