The Bluebirds' Matt Smith wrote his name into Chippenham folklore, scoring their first goal in the division after just four minutes.
Andrew Watkins quickly equalised for the hosts but David Pratt restored the visitors' advantage in the 16th minute, and strike partner Andy Sandell added another from the penalty spot before half-time.
William Richards put the game beyond doubt with little over 20 minutes remaining and then Sandell added the gloss, netting his second.
Bath's Connor Lemonheigh-Evans did get one back from the penalty spot late on, but it did little to ruin Chippenham's evening, who now have four points following an opening day stalemate with Havant and Waterlooville.
"It was a five-star performance!" manager Mark Collier told the club's website. "We had a game plan and we worked to it. We kept playing the ball forward as often as we could, we thought we could turn them and make use of the pace we have up front.
"We would like six points but we haven't. The next best thing is, if we can't win, we have to draw. We picked up a point on Saturday when I thought we were a little unfortunate not to get all three.
"On the road, you'd settle for a point at Bath but we've gone on the attack and really got at them. It was a squad performance, there's two or three sat in the stand too. We just have to keep on pushing.
Collier added: "The support from the fans was magnificent. When you see all those people behind the goal, in good voice, it's always great for the players."