"I had Bogut backpedaling, when I came off the pick-and-roll he was backpedaling," Evans said. "I just wanted to make a good move."
Bogut's baseline jumper hit the side of the backboard as time expired, allowing Sacramento to improve to 2-11 on the road.
Evans scored a game-high 24 points to top Milwaukee rookie Brandon Jennings, who had 15 points, nine assists and six rebounds.
"He got another shot at it and he is a great player," Jennings said. "You give great players second chances they are going to make something happen."
Jennings found Ersan Ilyasova under the basket for an easy basket with 5 seconds left, giving the Bucks a 95-94 lead. But that gave Evans enough time to win the game, and he took an inbound pass, lost Jennings on the screen, drove the lane around Bogut and hit the winning shot.
"Coach [Paul Westphal] told me it was my time to go out and make a play, and I did it," Evans said.
Jason Thompson scored 22 points -- including seven in the final three minutes -- and had 10 rebounds for the Kings, who were playing their fourth game in five nights.
Carlos Delfino scored 11 of his 17 points in the fourth quarter for the Bucks, who have lost three consecutive games and 11 of 14.
Milwaukee coach Scott Skiles said he was disappointed because he felt his team was not ready to play in the final moments of the game.
"We had blank looks on our faces and didn't appear in any way to have a whole lot of interest of what was going on out there," Skiles said.
Bogut missed a pair of free throws with 42 seconds left, and Luc Mbah a Moute missed one of two with 2:59 to go.
"Obviously, you need those to go down," Skiles said. "But I would not feel a whole lot different had we won the game. If we make a basket in the end and win that game I don't go home and break out the pom poms and start cheering."
Neither team led by more than seven the entire game. The lead changed hands seven times in the final two minutes, with the star rookies at the center of the action.
"We didn't feed into the 'Rookie of the Year' thing," Evans said. "We just went out there and played basketball."
Jennings was fouled with 1 second on the shot clock and made both free throws to give the Bucks a 91-90 lead with 1:05 to go.
Jennings grabbed a loose ball off the floor and was fouled at the basket, making both free throws to give Milwaukee a 93-92 edge with 20 seconds left.
Thompson took a pass from Evans and scored on a layup with 11.9 seconds remaining to give Sacramento a 94-93 lead.
"That was really a fun game for us," Westphal said. "It would've been a fun game for any neutral observer. I'm sure the Bucks didn't have that much fun at the end there."
Beno Udrih hit a running jumper with 2.1 seconds to go in the third quarter, giving the Kings their first lead of the second half, 73-72.
Evans already had 13 points on 5-of-10 shooting before Jennings took his first shot, a 27-foot 3-pointer that hit nothing but net and gave the Bucks a 43-42 lead with 3:22 left in the second quarter.
Bogut scored 13 points in the first half and the Bucks led 53-49 at halftime.
It was the first time Evans and Jennings played against each other since the 2008 McDonald's All-American high school game, also played in Milwaukee at the Bradley Center. They also played against each other in AAU games. ... Evans was born Sept. 19, 1989, four days earlier than Jennings. The only NBA player younger is Jrue Holiday, who is 19. ... Sacramento's only other road win was Nov. 7, a 104-99 victory at Utah.