The Eagles (20-6, 11-2 Big Sky) were leading 72-70 when Drew Brandon went to the foul line with 36 seconds left. He made the first free throw and when he missed the second went around the players on the lane and grabbed the ball on the baseline. He was fouled and made both shots to make it 75-70.
After a Southern Utah basket, Brandon made one of two and after a miss, Tyler Harvey made two to put the Eagles up 78-72 with nine seconds left.
Harvey, who is averaging 23.4 points a game, finished with 11 on 3 of 16 shooting.
Trey Kennedy led Southern Utah (7-17, 4-9) with 14.
Eastern, which went 20-8 as an independent in 1985-86, survived a drought of 8:20 without a field goal in the second half.