Tyreke Evans scored 22 points, DeMarcus 101.
Marcus Thornton finished with 15 points, including a pair of three-pointers during a late 18-4 run that put Sacramento on top for good.
The Kings scored 28 points off 23 turnovers to win for the fourth time in five games.
"We didn't do a good job of taking care of the basketball," Thunder head coach Scott Brooks said. "We have to clean that up."
Daequan Cook drained a three-pointer to put the Thunder on top, 95-87, with 5:56 remaining, and the visitors only made three field goals the rest of the way.
A Thornton three-pointer tied things at 97 with 2:07 to play, and Cousins went 1-for-2 at the foul line after grabbing an offensive rebound the next time down.
Westbrook hit a pullup jumper for a 99-98 advantage, but turned it over two of the next three possessions -- the first coming when he stepped on the sideline after Donte Greene hit a pair of free throws.
Evans added another free throw with 39.2 seconds showing, and following Durant's missed three, Cousins essentially put the game away with two more from the stripe.
The Kings led 23-19 heading to the second quarter, led 57-55 at halftime and trailed 78-76 entering the final stanza. There were nine ties and nine lead changes in the first three frames.
"The game never got too far away...and we won it down the stretch," Cousins said.
The Kings outscored the Thunder in the paint, 60-34...The Kings, who snapped a six-game slide against the Thunder, improved to 5-0 when scoring over 100 points...Jason Thompson posted 11 points and 10 rebounds for Sacramento, which will host the Suns on Saturday...Cousins was one rebound shy of his fourth straight double-double...James Harden logged 17 points for Oklahoma City, which visits the Jazz on Friday.