Paddy Jackson slotted over two penalties before full-back Olding burst through for the opening try to leave Ulster 11-3 ahead at half-time.
Olding squeezed over at the corner for his second try and replacement Declan Fitzpatrick added a late third.
Ulster move up to third in the Pro12 standings with the Dragons in 10th.
A downpour delayed the start by five minutes and the slippery conditions contributed to an error-strewn encounter.
Ulster pressed in the early minutes but they suffered a blow when captain Roger Wilson was forced off with a shoulder injury.
Dragons lock James Thomas got a touch ahead of Darren Cave in a race for Olding’s kick-through as the hosts narrowly missed out on a try.
The opening score came after 26 minutes with Jackson slotting over the first of two penalties in five minutes.
Cave’s precise lay-off sent Olding clear for the try with the promising full-back bustling through to the line, although Jackson failed to convert.
The Welsh side finally got on the scoreboard six minutes from the break thanks to an Angus O’Brien penalty.
Ulster’s strong start to the second half was rewarded with a second try after 47 minutes.
A flowing move concluded with Olding collecting Stuart McCloskey’s deft flicked pass before diving over in the corner, with the TMO judging he did not go into touch.
Jackson missed the conversion attempt from wide while O’Brien reduced the gap to 10 points on the hour mark with his second penalty.
The game deteriorated in the final quarter, with Ulster dominating, but misplaced passes and infringements saw the hosts miss out on securing a bonus point against a depleted Dragons side.
The third try came from the final move of the game, Fitzpatrick charging over from close range and Ian Humphreys adding the conversion.
