Centre Matt Banahan scored Bath's third try in their 36-15 success as Dragons slipped to a sixth successive defeat in all competitions.
Ross Wardle and Daniel Evans crossed for the Dragons, with Steffan Jones kicking the rest of their points.
Worcester produced an improved second-half performance to begin their campaign in style with a 34-18 victory over spirited Scarlets at Sixways.
Worcester trailed 13-6 at the interval but dominated the second half to score four tries, with Joe Carlisle kicking 14 points in all.
Scarlets battled to the end and were rewarded when Phillips won the race for the touchdown following a 60-metre kick and chase but Warriors had the final say with a last-minute try from Neil Best converted by Carlisle.
Ben Botica inspired Harlequins to claim a last-ditch 31-30 victory over Northampton in the Anglo-Welsh Cup at Franklin's Gardens.
The 23-year-old scored a try in injury-time before slotting the decisive conversion to snatch victory for Harlequins, who looked like they had blown a 21-9 half time lead.