It added up to a 106-92 victory Friday night.
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Odom finished with a season-high 26 points and 10 rebounds, and Bryant had 25 points, going 15 of 18 from the free throw line, hours after the defending NBA champions announced he signed a three-year contract extension worth nearly $90 million.
"I'm very excited to be here," Bryant said. "I knew I was going to be here."
The Lakers were prevented from clinching the Pacific Division after Phoenix won at Detroit. They were coming off a 2-3 road trip that ended with losses at New Orleans and Atlanta.
"The focus is to win and try to close up that spot so we don't have to put pressure on ourselves at the end of the season," Odom said.
Deron Williams had 20 points and 10 assists, and Carlos Boozer had 20 points and 18 rebounds for the Jazz, who had their three-game winning streak snapped. They had won eight of their previous 10, but the loss dropped them to fourth in the Western Conference behind the Lakers and Dallas and Phoenix, which are tied for second.
"Every time we needed to make a stop they would get a play, get an easy basket or a 3-point shot because they have the patience," Utah coach Jerry Sloan said. "They have the know-how. We just took shots."
Odom sparked the Lakers in the first quarter with 11 points, had two in the second and went scoreless in the third before launching the decisive run in the final 12 minutes.
"I was telling Kobe on the bench, 'I made shots I was supposed to make,"' Odom said. "Even on the road trip, I felt I was playing good defense and rebounding, but I wasn't making shots. I made a conscious effort to get to the hole."
The Jazz scored the first four points of the final quarter to close to 75-71. Odom promptly drilled a 3-pointer from the left corner and the Lakers regained command with an 18-3 run that restored their lead to 19 points.
Odom ran off nine in a row before Bryant scored five straight, then Odom capped the spurt with a putback dunk that increased the Lakers' lead to 93-74. Odom, Bryant and Gasol left the game with 3:03 to play and Los Angeles ahead 99-82.
"He always has good games against us for some reason," Williams said of Odom. "He's just one of those guys and we can't find an answer for him."
The Jazz twice cut their deficit to eight points in the third after the Lakers led by 14. Bryant missed 3 of 5 free throws and Gasol missed one before their lead dwindled to single digits.
The Lakers led by 19 in the second, but Utah outscored them 19-8 to close within eight on a 3-pointer by Wesley Matthews. Williams had 11 points in the spurt, including nine in a row. Odom's putback kept the Lakers ahead 54-45 at the break.
"He's great because he can make plays for himself and he can make plays for others," Boozer said.
Odom didn't miss a shot in the first quarter, hitting five field goals and a 3-pointer for 11 points in helping the Lakers build a 33-16 lead. They outscored the Jazz 20-2, getting nine by Odom and six by Bryant, and shot 64 percent.
- Los Angeles won the season series for the fourth consecutive year, going 3-1. The Lakers have won eight straight at home against Utah.
- The Jazz play three of their final five games on the road against non-playoff teams Houston, New Orleans and Golden State.
- Lakers F Luke Walton said he expects to return Sunday against San Antonio after being out with a pinched nerve in his back.