The 26-year-old scored a brace in Spartak's home win against Sevilla in last season's Champions League.
Promes helped fire Spartak to the Russian Premier League title in 2016-2017 and scored at least 12 league goals in each of his four full seasons at the club.
The versatile attacker has been included in the Netherlands squad for their Nations League opener on September 9 against World Cup winners France.
And Promes could be handed an immediate chance to make himself a Sevilla hero as his club head to Real Betis for a derby clash on Sunday.
Club president Jose Castro indicated Mariano had agreed to move to Sevilla before LaLiga giants Madrid moved in on the deal.