The Portugal international joined Milan from Porto last year as part of a significant recruitment drive by the Italian club.
But his debut campaign did not go to plan, scoring just twice in 24 Serie A appearances.
His haul of eight goals in 14 UEFA Europa League outings at least ensured his reputation did not take too much of a battering, as he earned a place in Portugal's FIFA World Cup squad.
But Milan has allowed him to depart the club on a temporary basis, with Sevilla finally signing a striker after missing out on a number of other targets.
Bas Dost, Vincent Aboubakar, Nikola Kalinic and Michy Batshuayi had all been strongly linked with the Andalusian side, but none arrived.
Silva, 22, will compete with Wissam Ben Yedder and Luis Muriel for a starting berth in attack, and Sevilla can buy the forward outright at the end of the season.