Sacramento hit just 1-of-6 shots from the field down the stretch and turned the ball over twice as well, giving the Timberwolves a chance to overcome what had been an eight-point deficit.
The lone basket by the Kings in the late going came from Evans with 41 seconds to go and pushed Sacramento in front by four.
Nikola Pekovic made 1-of-2 free throws for Minnesota with 34 seconds left and, after Evans suffered a turnover to give the Timberwolves an opportunity to force overtime, Dante Cunningham missed a 3-point try with 3 seconds left.
Isaiah Thomas led the Kings with 24 points and Evans added 21. Pekovic produced 18 points and 12 rebounds for Minnesota.