Ospreys almost home
But they were prevented from claiming the title because nearest rivals Munster kept the race alive with a 21-15 victory over Ulster.
Neath-Swansea opened up a 10-0 lead after just 13 minutes through a penalty from outside-half Shaun Connor and a converted try by wing Richard Mustoe, who out-sprinted the cover tackling to go over in the left corner. From then on the Dragons, courtesy of referee Orlan Trevor, had plenty of possession but could do little with it.