Norman Powell gave 15 points and Tony Parker had 11 points and six rebounds for the fourth-seeded Bruins (27-8), who defeated Arizona in the Pac-12 Tournament championship and are appearing in their 46th NCAA Tournament.
UCLA will take on Stephen F. Austin, which upset VCU in overtime earlier on Friday, in the round of 32 on Sunday.
D'Andre Wright had 18 points and James Woodard gave 11 points with nine rebounds for the 13th-seeded Golden Hurricane (21-13), who had won their last 11 games coming in and defeated Louisiana Tech in the title game to win the Conference USA Tournament.
UCLA had built its lead to 14 with just over nine minutes to play after Travis Wear slammed the ball home for a 57-43 advantage, but Tulsa came back with a 9-0 run, capped by two free throws from Wright that made it a 57-52 game with 5 1/2 minutes left.
That proved to be the Golden Hurricane's last gasp, though, as Parker converted a layup, Powell followed with a 3-point play, and Adams' 3-pointer with just under four minutes to play made it a 67-54 game.
Tulsa's Tim Peete followed with a layup, but that was the last basket that Tulsa scored until near the end of the game.
Powell's 3-pointer with seven minutes to play in the first half gave UCLA a 23-17 lead, and Parker had a layup and free throw a bit later for a nine-point advantage.
A 7-0 run later in the half saw Tulsa tie the game on a Woodard slam, but UCLA's Kyle Anderson followed with a 3-point play and Adams made two late free throws to give the Bruins a 35-30 advantage at the break.
A 10-0 run from UCLA early in the second half gave the team a 45-32 advantage.