USU (5-0) had the ball up five with 1:29 to go and almost ran out the shot clock before Shaw's dunk made it 73-66. Utah State then made a defensive stop to force Weber State to foul with 27 seconds remaining.
Preston Medlin missed the front end of a 1-and-1 and Davion Berry quickly scored for Weber State, but TeNale Roland answered with a wide-open layup and Spencer Butterfield made two insurance free throws to seal the Aggies' win.
“We just off to a good start," said Shaw. "They had a huge run in the first half. We were expecting the run, but we came back and fought tougher and had a good run in the second half. That helped us win.”
USU benefited from strong interior scoring. Along with Shaw's 20, Kyle Davis and Butterfield each finished with 16 points.
The Aggies are off to their first 5-0 start since 2008-09.
“This is how these games always are - hard-fought, comes down to a few plays," said Utah State head coach Stew Morrill. "We kind of got knocked on our keisters in the first half. We started out really well and then they just really got going and we couldn’t keep them contained. Usually both teams don’t play this good of offense because we know each other’s systems pretty well."
Berry led Weber State (0-3) with 21 points and the Wildcats shot 50 percent from the field.