Jeff Taffe scored two power-play goals for the Rampage, who were 4-for-10 with the man advantage. Yanick Lehoux and Kyle Wilson also scored power-play goals.
Bill Thomas also scored for the Rampage.
Duncan Milroy, Zack Stortini and Jonathan Ferland scored for the Bulldogs, who fell to 2-6-0-0 on the season.
Milroy opened the scoring at 18:36 of the first period. He broke in alone, pulled the puck to his backhand and beat Rampage netminder Philippe Sauve with a high shot to the top corner.
Thomas and Taffe scored two early second period goals to give San Antonio a 2-1 lead.
The Bulldogs responded with Stortini's goal at 12:29 of the second period. Just 13 seconds later, Ferland gave Hamilton a 3-2 lead.
Lehoux tied the game just over a minute later.
The Rampage scored the eventual game-winning goal, their third power-play goal of the game, when Taffe scored at 5:57 of the final period. Wilson added an insurance goal.
Sauve made 22 saves for San Antonio. Yann Danis stopped 26 shots for Hamilton.