Utah shooting guard Gordon Hayward scored a team-high 22 points and rookie point guard Trey Burke contributed 19 points, 10 rebounds and seven assists, but the short-handed Jazz (4-18) dropped their third straight.
Utah played without two of its usual starters. Power forward Derrick Favors missed his first game of the season with a sore lower back and forward Marvin Williams (sore heel) was sidelined for a third game in a row.
The Jazz took a seemingly safe lead at 91-84 with 2:59 remaining when Hayward split two free throws. But Cousins scored six points in the final three minutes and McLemore drained a wide-open 3-pointer to tie it a 97 with 3.2 seconds remaining.
Utah had a chance to win it in regulation, but Burke's 13-foot floater missed badly.
Thomas then outscored the Jazz by himself in overtime, 6-5, leading the Kings to their first win since a 113-106 victory at Phoenix on Nov. 20.
Backup guard Alec Burks scored 19 points for the Jazz, who only shot 40.4 percent for the game. Forward Jeremy Evans, who fouled out early in overtime, had eight points and a season-high nine rebounds in his first start this season.
NOTES: The Kings recalled center 7-foot C Hamady Ndiaye from their D-League affiliate, the Reno Bighorns. The Senegalese player saw action in nine games earlier this season for Sacramento. ... Ndiaye had three fouls in a 1:48 stretch against Jazz C Andris Biedrins, who went 1 of 6 from the free-throw line after being intentionally hacked. ... Sacramento was one of nine teams that Jazz coach Tyrone Corbin played for during his 16-year playing career from 1985 to 2001. Kings coach Michael Malone empathizes with Corbin, whose team is off to one of its worst starts in franchise history. "Tyrone Corbin's a friend, and I have a lot of respect for him as a coach and as a person," Malone said. "When you have your No. 1 pick out to start the season, that's never easy." ... Kings F Carl Landry and Jazz PG Diante Garrett both attended Milwaukee Vincent High School.