Doreena Campbell added 12 points and seven assists for the Bruins, and Markel Walker scored all eight of her points in the second half.
"Road wins are very precious," UCLA coach Nikki Caldwell said. "Any time you can get one, you take it."
It took a strong second half by the Bruins, especially on the defensive end, to pull this one out.
"I think it was mental," Dixon said. "We came back in (at halftime), talked to Coach Nikki, and she got in our heads. We came out with some fire. I think that was the difference in the second half."
After San Diego State's Jessika Bradley hit a jumper with 6:56 left for a 44-41 lead, UCLA held the Aztecs to one field goal until the final 8 seconds of the game.
"I didn't think we backed down, I wasn't disappointed in that," San Diego State coach Beth Burns said. "They exposed a lot of our youth and inexperience. You don't want a full-court game when only one person has been in that scenario. I thought they had a great game plan."
The Bruins, trailing 32-24 at halftime after shooting only 32.4 percent, outscored San Diego State 15-3 in the first 9 minutes of the second half. UCLA's pressure defense stifled the Aztecs, who committed 12 of their 19 turnovers in the final 20 minutes.
"Defensively, we started to shift our defense a little bit better and match up better," Caldwell said. "I thought we did a much better job of that and we extended our defense 94 feet and went to a little bit of pressure. I think that was disruptive as well."
Courtney Clements, a transfer from Arizona, scored 19 points in her first game at San Diego State, which was limited to 30.8 percent shooting from the floor.
UCLA won despite making just 11 of 25 free throws and committing 16 turnovers.
Walker scored six points in the first 8 minutes of the second half as the Bruins overcame their halftime deficit. UCLA improved to 45.8 percent shooting in the second half after struggling at the start.
"Walker did a good job," Caldwell said. "She is someone who is versatile, so we are playing her inside and outside."
The Bruins, who return four starters and six of their top seven scorers, have defeated San Diego State in 22 of 28 meetings.
Both teams played in the NCAA tournament last season, with San Diego State advancing to the round of 16 for the first time since the field was expanded to 64 teams.