The Men of Gwent trailed 10-7 at the interval after the boot of George Ford and a Rob Webber try cancelled out Dan Evans’ earlier effort.
A second-half stalemate lasted 20 minutes before the game burst into life. Italian debutant Kris Burton fired the hosts into the lead with a pair of penalties.
Bath responded through Anthony Watson but Jason Tovey edged his kicking dual with Tom Heathcote, who missed a last minute penalty.
Thomas Rhys Thomas and Matthew Screech made their debuts for the Gwent region, with the hooker also captaining the side.
While Jason Tovey pulled on the number 10 jersey following his switch from Cardiff Blues.
Following a scrappy and disjointed opening period the Dragons began to exert control of the encounter following a Tom Prydie break down the right .
Jack Dixon came close to bursting clear to score but following several phases and a scrum, Lewis Evans and Jonathan Evans sent Dan Evans over the whitewash.
Tovey added the extras, while George Ford was wayward minutes later. However Bath began to flex their muscle up front and a rampaging scrum allowed Ford to open their scoring.
The West Country giants took the lead shortly before the half-hour mark after Webber crashed over from close range with Ford converting.
Jones’ side pressed in the closing stages of the half, despite making mass changes, but the scores remained 10-7 at the break.
Replacement wing Anthony Watson came within inches of stretching Bath’s lead after bursting onto the ball and dancing through the Dragons’ midfield.
But he was held short and the Dragons survived unscathed. They then punished Bath for their lack of killer instinct when Burton levelled the scores shortly after the hour mark after the visitors were caught holding on at a ruck.
And the Italian fly-half fired the Dragons into the lead with a quick-fire second after Bath once again infringed at the breakdown.
But the Dragons’ hard work was quickly undone as Watson pounced on Wayne Evans’ loose pass to blaze home.
Tom Heathcote dissected the posts to hand the visitors a-four-point lead but Burton quickly cut it to one with his third penalty.
The Dragons were dealt a blow when Lewis Evans was sin-binned for insistent infringements at the breakdown.
But the numerical disadvantage made no difference as the Men of Gwent earned a scrum penalty, which Tovey duly slotted to regain the lead.
However, it once again lasted just minutes as Ian Nimmo was penalised and shown yellow at the breakdown, with Heathcote making the Dragons pay.
The Dragons appeared destined to fall to a slender defeat but the scores changed hands once again as Tovey nailed yet another penalty.
There was one more twist in a frantic finale as Bath earned a penalty at the death following a rampaging rolling maul, which saw Sam Parry become the third player sin-binned.
