Ulster, however, were pegged back just before the break when Aled Brew crossed for the Dragons, and Ceri Sweeney converted to draw the difference back to just one try.
Ulster pressed in the opening stages of the second period but were unable to capitalize and then had to withstand some late pressure from the visitors before they could guarantee taking the points and keeping just behind Leinster in the table.
Dragons Coach, Paul Turner, said: “On fifty minutes we’d silenced the crowd but we just didn’t take our chances. Ulster will probably feel themselves fortunate but we’ve come and challenged at the home of the league champions and deserved a draw in the end. It’s all part of our development and I’m really proud of the players, although perhaps we had too many unforced errors.”