The Bundesliga club had been in the market for a wide-man after Ibrahima Traore opted to join Borussia Monchengladbach on a four-year deal when his contract expired at the end of last season.
And, having previously earmarked Kostic as a transfer target, Stuttgart confirmed they had acquired the Serbia Under-21 international on a five-year contract.
"I had a good time in Groningen, but I now wanted absolutely to come to VfB Stuttgart," said Kostic. "Those responsible [for the transfer] have put much effort into me, we had good discussions and I'm glad that an agreement has now been reached.
"Now I am happy just to start the new task with Stuttgart and in the Bundesliga."
Kostic, 21, made 50 appearances in all competitions for Groningen, scoring 12 goals, and the fee for the player could climb by another €1 million.
The deal also includes a sell-on fee percentage for the Dutch club.
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