The NBA's No. 2 scorer, who entered the game averaging 27.7 points, missed four shots in the final two minutes.
"You try to make shots difficult," Hayward said. "You know he's going to get his shots off. He's going to get shots up. He's a very talented player.
"You just try to make sure every one is contested, challenged no easy ones. You try not to send him to the line, either. When he goes to the line a lot, that's when where he usually has his big games."
Durant finished with 18 points, but he scored only eight over the final three quarters.
Jazz coach Tyrone Corbin credited Hayward.
"He really did a good job of using his length [and] his quickness to stay between Durant and the basket," Corbin said. "He was mixing up the looks with him being up on his body some and confusing him."
Along with his teammates, Hayward helped close down the Thunder's effective pick-and-roll, especially down the stretch.
Oklahoma City shot only 6 of 21 in the fourth quarter.
"Instead of going under, [Hayward] mostly rolled him over the top," Corbin said. "He made Durant work before he got a look. I think he got a couple of open looks, but most of the shots he took were pretty contested."
Hayward did more than defend.
In the final minutes, he scored two points, handed out two assists, grabbed a rebound, came up with a steal and blocked a shot.
One of those assists led to one of the game's most critical basket.
After Oklahoma City narrowed Utah's lead to 87-86 with 2:42 remaining, Hayward threw a perfectly timed bounce pass to Paul Millsap from the left wing.
Millsap gathered the ball under the basket and dunked, stopping the Thunder's momentum.
Asked about the play, Hayward said, "The point guard is going to set a cross-screen. You read which way Paul is coming and which way the defender is going. …
"You try to put the ball where the defender is not. So I put it to Paul's outside hand, which happened to be along the baseline. He made a great catch and finish."
A closer look
Kevin Durant's fourth quarter Tuesday:
Minutes FGs 3s FTs Reb Ast Pts
8:12 0-7 0-3 0-0 2 0 0