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Duncan scored a season-high 38 points and grabbed 11 rebounds, but had little help down the stretch after David Robinson fouled out with 1:17 remaining.
Duncan dominated the final minutes of regulation, completing a three-point play with 1:06 left before sending the game to overtime on a bank shot with 13 seconds left.
He scored the first basket of the extra session, but the Spurs missed their next five shots and committed a 24-second violation while Sacramento pulled away.
Turkoglu nailed a running bank shot to forge a 99-99 tie with 3:25 left in overtime, and Bobby Jackson made a layup to give the Kings the lead for good with 2:40 left.
Turkoglu made three free throws in the final 2:01 and Peja Stojakovic added a pair while San Antonio struggled to score and fell to 7-1 at the Alamodome.
"Down the stretch, our execution was not as sharp as it should be," San Antonio coach Gregg Popovich said. "Our offensive execution in overtime left a lot to be desired. We have to work on that."
"We had opportunities to do it and we took it away from ourselves," Duncan said. "We made mistakes and they capitalized and we didn't."
Lawrence Funderburke scored a season-high 16 points off the bench and Stojakovic finished with 11 and eight rebounds for the Kings.
"Our guys have just been playing with a new determination," Sacramento coach Rick Adelman said. "They understand that other players have to step up. They just believe that they can. It doesn't matter who we play, they just believe they can win and that attitude has rubbed off on me."
Antonio Daniels scored 14 points for San Antonio, and Robinson had 10 and 12 rebounds.
The Spurs shot 44 percent (39-of-88) and made 26-of-18 free throws but went to the line just once after the third quarter.
Sacramento made 28-of-37 from the line, including 18-of-23 in the fourth quarter and overtime.
The game was tied at 47-47 late in the second quarter when the Kings went on a 17-9 spurt to take a 64-56 lead with with 7:56 left in the third. Stojakovic scored seven points in the run and Funderburke added four, capping it with a dunk.
Sacramento had a 71-64 lead before the Spurs closed the third period with a 17-4 run, highlighted by eight points from Duncan and seven from Daniels.
San Antonio held a 85-77 advantage with 10:10 left in regulation, but the Knigs mounted a 15-5 run and tied it on a pair of free throws by Bibby with 2:40 to play.
Bibby nailed a jumper and Turkoglu sank two free throws to make it 96-92 before Duncan was fouled dunking over Vlade Divac and converted the free throw.
Turkoglu made 1-of-2 free throws with 17 seconds left, and Duncan forged the final tie of regulation on the next possession, driving past Divac and banking the ball off the glass with his right hand.
"Sometimes when you have a chance to win a game in regulation and you fail to do that, you have a let down," Adelman said. "I was proud of the way our guys were able to sustain their effort and see this game through."