With the game tied at 28, Golden pulled in a 10-yard scoring pass from Showers with 3:44 remaining. With 2:03 left Showers found Golden on a 32-yard scoring strike for a 42-28 lead.
UTEP (3-3, 1-1 Conference USA) then ran out the clock for the win.
The Miners had a 21-7 lead at the half, but were outscored 21-7 in the third. UTEP has been outscored 83-20 in that quarter in six games.
Showers completed 17 of 26 pass attempts for 152 yards while Aaron Jones rushed for 103 yards.
Heinicke led Old Dominion (3-4, 1-3), throwing for 196 yards and three touchdowns. Ray Lawry rushed for 121 yards and two scores.