Martin had 13 points, including a free throw with 23 seconds left. Dougher added another free throw with 6 seconds left in the game, which started at 6 a.m. EDT.
Martin's free throw gave Stoney Brook a 50-49 lead. After Monmouth turned the ball over on the next possession, Dougher also made one of two from the line for the Seawolves (2-1).
R.J. Rutledge's 3-pointer for Monmouth (0-1) was hard off the backboard at the buzzer.
Dougher said getting up early didn't bother him.
"Once the game starts and the ball goes up, it's just like a 7 p.m. game," Dougher said of the nationally televised contest. "We had 6 a.m. practices to get ready for this one. We were ready. It didn't have an effect."
Ed Waite had 13 boards for the Hawks.
"It wasn't a pretty game," Stony Brook coach Steve Pikiell said. "We gutted out a win at a very tough place to play. We executed down the stretch and grinded out a good win. We had some guys make big plays."
Monmouth led 24-20 at halftime. The Hawks extended the lead to 34-23 in the first five minutes of the second half, with Campbell scoring on a putback and hitting a 3-pointer and a basket by Marcus Ware.
The Seawolves went on a 9-0 run to get to within 34-32, with Al Rapier scoring on consecutive drives. But the Hawks scored five straight points to take a 39-32 lead on a 3-point play from Rutledge with 11:05 left.
The Seawolves tied the game on a 3-point play by Anthony Mayo with 4:06 remaining. Martin made a 3-pointer with 1:38 left that gave Stony Brook its first lead, 47-46, since early in the game.
Rutledge put Monmouth ahead at 49-47 with 1:11 left on a 3-point play, following his own miss and making the free throw. But it was the last points the Hawks would score.
Stony Brook shot 35 percent, while Monmouth made only 30 percent of its shots. Stony Brook went just 11 of 20 from the line.
Last year, Monmouth also lost an early tip to St. Peter's College.