"It felt good to finally get one of those wins, and we can move on now," Durant said.
Westbrook connected on a 3-pointer from the left wing to open the scoring in overtime, and Durant rattled one in from about the same spot two possessions later for a 108-104 advantage.
Westbrook then answered DeAndre Jordan's three-point play with a baseline jumper, and Oklahoma City was able to close it out from the foul line.
"This is a good win for us, especially against a team like that, that had some success lately," said Thabo Sefolosha, who hounded Clippers All-Star Chris Paul much of the game.
Blake Griffin led Los Angeles with 23 points and Jamal Crawford scored 20 for the Clippers, who had a six-game winning streak snapped after entering the game tied for the best record in the West.
After trailing for the previous 44 minutes, the Clippers finally pulled even at 102 when Matt Barnes drove for a layup with 36 seconds left in regulation. Griffin denied Durant the ball on the Thunder's ensuing possession, and Westbrook came up empty on a drive to the basket.
Paul then dribbled out the rest of the clock before missing a turnaround jumper from the left elbow as the horn sounded.
"I think the frustrating part is we had an opportunity to win and, regardless of what anybody says, it's going to be hard for us to win a game when I play that bad," said Paul, who missed 12 of 14 shots and ended up with nine points and nine assists.
"It's tough because you work so hard but there's going to be nights like that."
Los Angeles had been tied with Memphis for the West's best record at the start of the day but dropped behind Oklahoma City and San Antonio with the loss. The Clippers were trying to win at least seven games in a row for only the sixth time in franchise history, and the first time since 1991.
"We fought the whole game and Chris got a great look. We're going to live and die on that shot," Jordan said. "We fought but we definitely made some mistakes tonight. They made more plays than we did down the stretch."
Matt Barnes scored 19 points while replacing starting forward Caron Butler in the Clippers' lineup. Butler missed the game with a strained right shoulder sustained Monday night in a win at San Antonio.
Kevin Martin scored 20 points and Serge Ibaka chipped in 15 points and 12 rebounds for Oklahoma City before fouling out in overtime. Hasheem Thabeet also matched his career high with 10 points off the bench.
The Clippers won three out of four against Oklahoma City in a sometimes testy series last season that included Griffin's memorable throwdown over Perkins in a Clippers win and Perkins' retaliation with a hard foul to Griffin's face as Oklahoma City won the rematch.
Barnes had a couple scuffles with Ibaka and Perkins but nothing quite as noteworthy.
The Clippers ended up with a 54-38 edge scoring in the paint and a 28-10 advantage in second-chance points. Oklahoma City made up for it by making 34 of 39 free throws, while L.A. was 24 for 33.
Durant was 19 for 21 at the line, with coach Vinny Del Negro telling official Kevin Fehr after one call: "He doesn't need any help scoring."
"Those guys are physical and (I was) just trying to fight through it. Just trying to get there, man," Durant said. "I think early on, me driving and me getting into the paint, I was a little hesitant in the second half because I was turning the ball over.
"But I just tried to keep going. My teammates told me to just keep being aggressive, and that's what I tried to do."
The Thunder pushed ahead with a string of eight straight made baskets in the first quarter, and the lead stretched to 26-16 after Westbrook's extra effort turned a potential turnover into a three-point play for Thabeet.
The Clippers finally got back within striking distance early in the fourth, with Thunder coach Scott Brooks yelling to Durant to get off the bench with Los Angeles back within 78-75.
With Durant back in, Oklahoma City responded with the next six points to stabilize momentarily before the Clippers' final charge.
Los Angeles has held eight straight opponents below 46 percent shooting. The Thunder were just shy at 45.6 percent. ... Crawford made all four of his free throws, stretching his streak of consecutive makes to 29. He made 37 in a row in April. ... Jordan, one of the NBA's worst foul shooters at 42 percent, got Oklahoma City to stop hacking him intentionally in the fourth quarter by making three of four.