James Walshaw scored a hat-trick for the Seadogs in a strong display at the Flamingo Land Stadium, while Michael Coulson and Ryan Watson also bagged goals.
"I thought the boys were excellent," manager Darren Kelly told Scarborough Athletic FC Media.
"They needed to be. The conditions, we played against the wind first half and I thought we were fantastic.
"Going in half-time 2-0 up, could have been more, but to be fair to Bamber Bridge you can see the way they work on a lot of stuff. A good footballing side and we needed to be on top of our game.
"I thought how we passed the ball, the intensity levels, how we broke lines, how we linked play up and how midfielder and strikers linked was excellent.
"I thought how we worked out of possession was brilliant. We missed a penalty, it could have been more. I thought the boys were fantastic."
Scarborough now sit just outside the play-offs in sixth position.
South Shields bounced back from their league defeat last weekend to claim a 3-1 victory over Grantham Town.
Lee Mason opened the scoring inside 12 minutes, before Lee Hughes equalised just before the break.
However, second-half goals from Dominic McHale and Joshua Gillies saw the Mariners come away with all three points.
Whitby Town defeated Stafford 2-1 with goals from Matthew Tymon and a penalty converted by Paul Blackett.
Stalybridge Celtic also claimed the three points away from home, defeating Gainsborough Trinity 3-1. Goals from Patrick Lacey, Berat Ustabasi and Scott Bakkor.