Brian McLaughlin's side only pulled clear in the final quarter as their extra strength and experience saw them through against a brave challenge from the Scots who simply ran out of steam.
Indeed, only some last-ditch defending from Sean Lineen's men prevented Ulster from grabbing a bonus point.
Tries from Darren Cave, Paul Marshall and skipper Chris Henry saw Ulster clear, while Ian Humphreys kicked 13 points.
It was by no means all one-way traffic as the visitors opened the scoring after 10 minutes when Troy Nathan barged over the line near the posts, shortly after Colin Shaw was held up in the left corner. Referee Leighton Hodges went upstairs to the television official and the score was given, but Duncan Weir was wide with the straightforward-looking conversion.
Ulster were stung into action and responded with a Humphreys penalty before Cave snaffled a try following Ian Whitten's mazy run when the Ulster centre intercepted a pass from Colin Gregor to Stuart Hogg. Humphreys added the easy extras and Ulster led 10-5 after 20 minutes.
The remainder of the half saw neither side carve out a concrete scoring chance but then a careless slap-down of the ball from Lewis Stevenson allowed Weir to slot his first kick with a 40th-minute penalty to cut Ulster's lead to 10-8.
Ulster again responded and after the impressive Nevin Spence had won the restart back for the home side, the ball was moved right and, from a Dan Tuohy drive, Marshall nipped over in the right corner for a well-worked score which Humphreys was unable to convert.
That should have seen Ulster through to the break but four minutes over the 40-minute mark, Weir goaled a long-range penalty after Ulster failed to roll away in the tackle and the half ended with Ulster's lead cut to 15-11.
The Warriors came tearing out of the traps in the second half and Ulster were fortunate to escape with only a penalty under their posts when Weir narrowed their lead to one point in the 47th minute.
Still, Humphreys immediately struck a long-range reply to put his side 18-14 up and then, just before the hour mark, landed a penalty from halfway to give Ulster some breathing space.
After that, Ulster again resisted some serious pressure from the Warriors but finished the stronger, although Humphreys was wide with a 73rd minute penalty.
Still, they got their reward for so much late pressure when Henry barged over near the posts in the 78th minute with Humphreys adding the extras and putting Ulster well clear.
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