The 26-year-old has signed a four-year contract that will keep him in Russia until the summer of 2017.
Reports claim Spartak have paid around £3.6million for Tasci.
“This has not been an easy decision and it’s definitely no decision against VfB Stuttgart,” he told the club’s official site.
“It’s a decision of a new task. After so many years with VfB Stuttgart I like to face a new challenge in my professional career.”