The four massive Galilean satellites are believed to have formed within a circumplanetary disk during the last stages of Jupiter's formation. While the existence of a circum-Jovian disk is supported by hydrodynamic simulations, no consensus exists regarding the origin and delivery mechanisms of the building blocks of the forming satellites. The opening of a gap in the circumsolar disk would have efficiently isolated Jupiter from the main sources of solid material. However, a reservoir of planetesimals should have existed at the outer edge of Jupiter's gap, where solids were trapped and accumulated over time. Here we show that the formation of Saturn's core within this reservoir, or its prompt inward migration, allows planetesimals to be redistributed from this reservoir toward Jupiter and the inner Solar System, thereby providing enough material to form the Galilean satellites and to populate the Main Belt with primitive asteroids. We find that the orbit of planetesimals captured within the circum-Jovian disk are circularized through friction with gas in a compact system comparable to the current radial extent of the Galilean satellites. The decisive role of Saturn in the delivery mechanism has strong implications for the occurrence of massive moons around extrasolar giant planets as they would preferentially form around planets within multiple planet systems.