Eversley tipped in a miss for a 67-65 lead and Cal Poly didn't trail the rest of the way.
Odister made free throws for a 68-65 lead and 70-68 edge with 14.2 seconds remaining.
Jordan Adams missed a 3-point attempt and time expired for the Bruins (4-2), who needed overtime to defeat another Big West opponent, UC Irvine, earlier this month.
UCLA made only one field goal in the final three minutes, on Adams' layup underneath to tie it 68.
Freshman guard Shabazz Muhammad led UCLA with 15 points and 10 rebounds in his debut at Pauley Pavilion. Travis Wear had 14 points and Adams scored 13.
No. 23 Colorado 89, Air Force 74 • In Boulder, Colo., freshman Josh Scott helped break open a tight game by scoring 13 of his 20 points in the second half, and Colorado (5-0) handed Air Force (5-1) its first loss of the season.
No. 17 Gonzaga 81, Davidson 67 • In Lake Buena Vista, Fla., Elias Harris scored 24 points, Kevin Pangos added 23 and Gonzaga beat Davidson in the Old Spice Classic championship game.
Kelly Olynyk added 14 points for Gonzaga (6-0).