The Vikings entered the contest 0-11 against Top 25 teams since reinstating their basketball program for the 1996-97 season, and hadn't won in Spokane since 1974.
Sophomore Phil Nelson hit the deciding shot for the Vikings with 1:09 left in the game.
With the score tied at 69-69, Nelson, who had missed his first five 3-point attempts, sank a long three and Dominic Waters followed with two free throws with 30 seconds left to secure the win.
Gonzaga missed all three of its field goal attempts over the final minute and the Vikings celebrated after the Bulldogs Jeremy Pargo missed a 3-pointer as time expired.
Portland State led 67-62 with 3:41 to go after a layup by Julius Thomas, but Gonzaga rallied to eventually take the lead.
A three by Pargo put the Bulldogs up 69-67 with 2:16 remaining before Waters hit a jumper to tie the score.
Portland State won the battle on the boards, tallying 39 rebounds to Gonzaga's 30.
The Vikings shot 45.9 percent (28 of 61) for the game and went 12 of 31 from behind the arc.
Gonzaga finished with a 42.9 (24 of 56) shooting percentage.
The Bulldogs held an early 6-4 lead after a dunk by Josh Heytvelt, but PSU came back and tied it up on a dunk by Nelson that ignited a 11-0 Vikings run.
Dominguez followed Nelson's dunk by hitting three straight 3-pointers and PSU was up 15-6 at the 14:17 mark of the first half.
A 3-pointer by Matt Bouldin knotted the score back up at 15-15 and the Bulldogs later went up 24-21 with 8:54 to go on a 3-pointer by Steven Gray.
Portland State's Andre Murray answered with a three of his own to tie the score back up at 24-24 and PSU scored nine straight points from there to go up 33-24 with 5:28 remaining until halftime.
Gonzaga (8-3) closed the first half on a 13-5 run to close within one point at the break, 38-37.
Dominguez shot 8 of 12 from the floor, including 7 of 10 from 3-point range, and also led the Viks with six rebounds and five assists.
Nelson and Waters each added 16 points. Nelson also chipped in five rebounds and Waters had two steals.
Bouldin paced Gonzaga with 18 points, while Heytvelt tallied 13 points and eight rebounds.
Austin Daye finished with 12 points and nine rebounds for the Bulldogs.
After tough losses at Washington and Cal Poly, the Vikings have won two straight and are now 9-3.
The win was especially rewarding for PSU after the tough time they had getting to Spokane. Their flight out of Portland on Sunday was cancelled and they eventually took a charter bus to Bellevue, Wash., yesterday. They finished the drive to Spokane today and got to the arena just a few hours before tip-off.
Portland State returns home for Christmas before flying to Texas to to take on Texas Southern on Saturday at 5 p.m. PT. They then face 21st-ranked Baylor on Monday.