The eight-time Super Rugby champions had raced out to a 26-0 lead inside the first 20 minutes but the visitors dominated the next 40 to reduce the deficit to 31-21 and put pressure back on the home side.
The Crusaders, led by captain Sam Whitelock and the rest of his pack, wrestled back control of the game in the final quarter to blow the scoreline out as the visitors made too many errors.
The Stormers, who have not won in New Zealand since 2013, were unable to get anything going in the first quarter as their set piece failed to function.
The home side also seized on numerous turnovers to score tries to Bridge, hooker Codie Taylor, flanker Billy Harmon and scrumhalf Bryn Hall.
Hall's effort came just seconds after the Stormers had restarted following the try by Harmon, who was making his first Super Rugby start after Matt Todd was ruled out with a broken thumb.
Facing a 26-0 deficit and with 60 minutes to play, the Stormers finally got into the game and slowed the onslaught before they gaining some reward for their control inside the home team's half with prop JC Van Rensburg crossing.
Bridge, however, scored his second when the Crusaders again swept down field to give his side a 31-7 lead at the break.
The Stormers, however, were a different side in the second half and it took less than 60 seconds for them to kickstart a comeback with prop Wilco Louw barrelling over before lock JD Schickerling crossed to cut the lead to just 10 points.
The Crusaders tightened up, began to play territory and control the ball with a decision by Whitelock to go for attacking lineouts rather than kickable penalties allowing Richie Mo'unga to create space for Seta Tamanivalu to cross.
Mo'unga added his own try shortly after, before Springboks prop Steven Kitsoff crossed in the final five minutes.