Wesley Skiffington's extra point provided the winning margin for the Seawolves (1-2), who drove 73 yards in 14 plays for the final score.
Coulter was 10-for-12 for 137 yards and the touchdown to Porter. Dayne Hoffman also threw for 59 yards and a score for Stony Brook.
Kyle Newhall-Caballero was 29-for-45 for 267 yards and three touchdowns for Brown (0-1), the defending Ivy League champion.
Newhall-Caballero threw a 20-yard touchdown pass to Matthew Sudfeld, a 23-yarder to Buddy Farnham and a 6-yarder to Alex Prestley, which gave Brown a 20-14 lead with 5:36 left in the third quarter. However, Drew Plichta missed the extra point.
Farnham caught 12 passes for 121 yards.