The boot of Dan Jones and a try from Hadleigh Parkes had opened up a lead but Rory Scholes' score and late penalty try turned the tide back in Ulster's favour.
And with a late Aled Thomas penalty denying Ulster even a losing bonus point, there was much for Pivac to cheer despite Peter Edwards, John Barclay and George Earle all collecting late yellow cards.
"I never wanted to win one of those (fair play awards) but there is a balancing act isn't there," said Pivac.
"At 19-12 with a few minutes to go and yellow cards going on it was very similar to last season (when Ulster fought back to draw 32-32), but luckily for us we were able to control those final moments at the right end of the field and that penalty from Aled Thomas was important to deny them a losing bonus point.
"We started well like last week, and we had the bulk of possession and territory in that first 20-odd minutes.
"We took our foot off the throat a little towards the end of the first half and the penalties started coming, but they missed a shot at goal and we went in 9-0 up.
"We started the second half well and after that were able to stem the tide at the right times to get away with that win.
"When we are down in numbers we were able to fight hard for each other and really dig in because last year we were giving those leads away.
"That is pleasing, but we would like to be in a position by round three where we tidy that discipline issue up."
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