Sweden had only made it into the Round of 16 for the first time thanks to already-eliminated Great Britain pulling off a surprise to beat Montenegro, which saw the Blagult progress in fourth place on points difference.
Turkey, meanwhile, finished top of Group A with five wins from five, including a victory over Olympic silver medallists Serbia, and had the added advantage of not having to leave their base in Riga, while Sweden made the trip from Tampere, Finland.
But the tone for a potential upset was set right from the off, as Sweden went 6-0 and then 9-3 ahead in the opening minutes, with Ludvig Hakanson taking no time to find his range with five of those points.
Though Turkey levelled things up at 11-11, the Swedes never fell behind in the first 10 minutes, as Melwin Pantzar and Pelle Larsson contributed to grow the lead back to six, at 21-15, before ultimately leading 23-20 at the end of one.
The dream of an upset looked to be fading for Mikko Riipinen's men early in the second quarter when Turkey tied things up again at 28-28, as Alperen Sengun and Sehmus Hazer grew into the game.
But three points from Pantzar set Sweden on their way to going on an 11-4 run, culminating in Simon Birgander's jump shot handing them their biggest lead of the game at 39-32. The chance of a real scalp looked on at the break, as three from Denzell Andersson made the half-time score 42-37.
Early in the second half, the "12 Giant Men" found the form they have been showing in Latvia for the past two weeks; Ercan Osmani scored seven points as Turkey went on an incredible 14-0 run to turn the match on its head, turning a seven-point deficit into a seven-point lead, 51-44.
Adam Ramstedt finally ended the streak, but Ergin Ataman's side were unrelenting, and Adem Bona's layup put them 61-50 up with just over a minute of the third quarter to play.
Hakanson and Viktor Gaddefors added some respectability to the score to make it 63-55 with 10 minutes to play.
Turkey were doing enough to keep the Blagult at bay in the first exchanges of the final quarter, before the underdogs produced a superb late rally - capped off by a Hakanson three-pointer, Sweden put together an 8-0 run to turn 69-61 into 69-69 with 06:12 left to play.
But crucially, they couldn't find the go-ahead score, and when Sengun added two to make it 82-76 in the final minute, Turkey had done enough to book a place in the last eight, taking an 85-79 victory.
Sengun of the Houston Rockets top scored with 24 points, led the game in rebounds with 16 and jointly topped the assist list with Pantzar on six. Hakanson's 16 points were the best for Sweden, as Larsson and Gaddefors managed 15, one more than Simon Birgander, who had 13 rebounds.