UCSB (2-7), which lost its fifth straight, was led by Gabe Vincent's 16 points.
UCSB was nine back with three minutes left but the Zips used 10 free throws and a basket by Robotham to ensure the win.
The Zips came in as the nation's second-ranked team in 3-point defense (23.3 percent) and third in overall field-goal defense (34.8). But the Gauchos shot 46.4 percent, including 35 percent from 3, in reaching 70 points for the first time this season. But they also had 19 turnovers and got just two second-chance points off 10 offensive rebounds.