Former England centre Matt Banahan scored Bath's third try as Dragons slipped to a sixth successive defeat in all competitions. They had also conceded 46 points to Bath when the sides met in the LV= Cup last season.
With 14 on the casualty list, including Wales flanker Dan Lydiate, they named three teenagers in the starting line-up, including 17-year-old Jack Dixon at centre.
Heathcote opened the scoring with a 50-metre penalty, but the Dragons responded immediately by outflanking the Bath defence and left wing Ross Wardle appeared inside full-back Hallam Amos to score. Steffan Jones added the conversion.
A frenetic first 10 minutes continued with a try for Rokoduguni, supporting Heathcote's break up the middle after Bath won back possession on the right touchline.
The fly-half then chipped over the conversion to give the home side a 10-7 lead.
Rokoduguni, who represented the Army while serving in the Royal Scots Dragoon Guards, was alert in defence too, chopping down Pat Leach when the centre pierced the Bath defensive line.
A sequence of penalties for the fly-halves saw Bath stretch their lead to 16-10 as the teams continued to play at pace, but at the cost of infringements and handling errors.
Bath lost both scrum-half Mark McMillan and centre Dan Hipkiss to injuries and the Dragons had to replace Amos, one of their more effective runners.
Heathcote landed two more penalties, both from awkward angles, the second being awarded on the stroke of half-time for a late and high tackle on the Bath number 10 by Cornish lock Ian Nimmo, who was sin-binned.
Bath wasted no time after the break in making the numerical advantage count. A multi-phase attack ended in Banahan crossing under the posts after a pop-pass by Josh Ovens. Heathcote converted.
When Newport eventually established a foothold near the Bath line, winning a line-out in the corner, Hugh Gustafson's throw to Nimmo was crooked.
They kept up the pressure, though, and Wardle looked to have crossed in the opposite corner but had put a foot in touch.
Bath were then able to work their way out of trouble after Stuart Hooper stole a line-out.
The line-out continued to be a problem area as skipper Jevon Groves was yellow-carded for taking out the Bath jumper, Ovens. It was just one of a string of penalties conceded as the Welsh side resorted to desperate defence.
The inevitable third try came on 72 minutes when Rokoduguni timed his run perfectly to take Heathcote's pass and score under the posts.
Heathcote converted, but the Dragons roused themselves to create one final unconverted try for replacement wing Dan Evans.
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