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The Bianconeri started the game four places below their hosts, but Kostic struck in the first half to send them on their way to completing the league double over Inter for the first time since the 2019-20 campaign.
Simone Inzaghi's side mustered just three shots on target as Juve's backline saw out the win with relative ease.
Inter's patchy form continues, with a third defeat in four Serie A games seeing their Champions League hopes suffer yet another blow as Massimiliano Allegri's men move within nine points of the Nerazzurri.
After Wojciech Szczesny twice denied Nicolo Barella early on, Kostic brilliantly put Juve ahead with 23 minutes played.
The Serbia international received the ball from Adrien Rabiot before rifling into the bottom right corner, with a lengthy VAR review for a suspected handball in the build-up eventually finding no wrongdoing.
Romelu Lukaku spurned a chance to equalise when he headed wide before the break, before Dusan Vlahovic and Manuel Locatelli both tested Andre Onana early into the second half as Juve looked to build their advantage.
Allegri introduced Federico Chiesa off the bench, and the substitute nearly made it 2-0 when his powerful run ended with him curling just wide.
Inter pressed for a leveller late on, but Juve's resilient defence stood firm to see out the remaining minutes for another clean sheet and make it eight wins from 12 in Serie A since the turn of the year.