The 26-year-old, who joined Spartak in 2014, helped the Russian club win its first Premier League title in 16 years in 2017.
He scored 66 goals in 135 matches during his spell with the club.
"It's time for a new chapter in my journey," Promes wrote on Twitter.
Promes, who underwent a medical with Sevilla earlier on Friday, has won 28 caps for the Netherlands.
Sevilla and Spartak did not disclose the terms of the deal.
Sevilla finished seventh in La Liga last season, qualifying for Europa League.
(Reporting by Gabrielle Tétrault-Farber; Editing by Toby Davis)
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