Dombrowski, who split time with sophomore quarterback Buck Pierce, ran for three touchdowns, including two in the game's final 18 minutes.
Dombrowski's last score came on an 11-yard option play with 3:28 remaining.
Idaho marched downfield on its next possession, but the Vandals came up empty when Aggies defensive back Siddeeq Shabazz intercepted Michael Harrington's pass at the New Mexico State 19 with 1:45 left.
The Aggies (7-5, 5-1 SBC) ran out the clock, then celebrated the school's first seven-win season since 1967.
Dombrowski, whose third-quarter touchdown run cut Idaho's lead to 31-28, was 9-of-13 passing and had 20 carries. Pierce was 10-of-17 for 158 yards.
Harrington, who was scheduled to alternate quarters with starter Adam Mallette, took over for good after going 6-for-6 and leading the Vandals (2-10, 1-5) to a touchdown on his opening drive.
He threw for another score with 20 seconds left in the first half to put Idaho ahead 24-14 at the break. The Aggies closed to 24-20 after a pair of third-quarter field goals by Dario Aguiniga, but Idaho answered with a 39-yard touchdown run by Blair Lewis.
Dombrowski scored on a 21-yard run to bring the Aggies within three at 31-28. It was the third fourth-quarter comeback of the year for New Mexico State and the sixth second-half comeback.
Harrington, the younger brother of Detroit Lions quarterback Joey Harrington, was 20-of-35 passing for 219 yards. Lewis had 159 rushing yards on 20 carries.