However it was the Warriors who made the better start with former Connacht man Nathan barging through the tackle of Ian Humphreys to reward the visitors for a period of concerted pressure on ten minutes, Duncan Weir missing a straightforward conversion.
But after Humphreys had got Ulster on to the scoreboard six minutes later with a simple penalty, Colin Gregor threw a wild pass which Cave collected to streak under the posts from 30 metres out, to make the score 10-5.
The home side continued to pile on the pressure but it was Glasgow who scored next through Weir after Lewis Stevenson was penalised for knocking the ball out of Gregor's hands.
Ulster scrum-half Paul Marshall immediately took advantage of a great charge from Dan Tuohy to dart over from close range three minutes before the breal but Weir slotted a second long-range penalty to make it 15-11 at half-time.
As in the first half, Glasgow made the better start in the second, but after another penalty from Weir brought them back into it, Colin Shaw gave away a penalty for Humphreys to restore the four point lead.
Humphreys added another soon after to widen the gap before Henry charged over off the back of a five scrum on 75 minutes after Ulster had been held up just short by a valiant Glasgow defence.
With the result no longer in doubt, Ulster poured forward in search of a bonus point try and looked set to pick up five points with the Glasgow defence stretched.
But Henry spilled a ball with an overlap on, and the Warriors saw out the remaining minutes.
Ulster will now turn their attentions to next week's trip to Aironi while Glasgow will try to bounce back against reigning champions Munster.
|Ulster Rugby Score Card|
|Glasgow Warriors Score Card|