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.”
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.