Eric Uba scored two goals in the first period as the Storm skated to a 3-0 lead before the game was 14 minutes old at Sleeman Centre.
But the Bulldogs battled back in a big way after Zachary Roy replaced Marco Costantini in net after the latter allowed three goals on just nine shots.
Avery Hayes scored his fourteenth goal of the season with seven seconds left in the first period and Tag Bertuzzi and Isaac Nurse added their seventeenth and fourth goals of the season, respectively, in the second period to even the game 3-all.
The Storm regained the lead with 6:49 to play in the third period when Cedric Ralph converted a one-timer past Roy for his thirteenth goal of the season and a 4-3 lead.
Hamilton next plays Tuesday at 4 p.m. when they host the Mississauga Steelheads at FirstOntario Centre.