The Scarlets led their visitors by 12 points heading into the final five minutes courtesy of a Hadleigh Parkes try and the metronomic boot of Steve Shingler.
But scores in the final moments by Felix Jones and JJ Hanrahan, following on from Jack O'Donoghue's earlier effort, allowed Munster fly-half Ian Keatley to level the scores at 25-25 with the game's last kick.
That late capitulation, which didn't prevent the Welsh side from leapfrogging Connacht into the PRO12 top six, left Pivac in disconsolate mood despite a strong start.
"We worked our way into a winning position at 25-13 and then we were the authors of our own demise," he told Wales Online.
"There were some little turnovers from us and that allowed them into our territory in attack, and all of a sudden they are in the corner.
"I am so disappointed. We worked hard in that first half defensively with very little possession.
"Going into half-time at 12-6, we were in a good position.
"We just had to get our hands on the ball and play a bit more rugby. We talked about not kicking it away so much when we got around halfway and hold onto possession."
With Connacht still to play this weekend, it is shaping up to be a three-way battle between the Irish province, Pivac's men and Edinburgh Rugby to break into the PRO12 top six.
And following their third draw of the campaign, the Scarlets head coach suggests the club's recent form is showing promise.
"There are six points that have gone begging, but this was the most frustrating," Pivac added.
"We have said this is a big month for us when we are playing all of the four Irish teams.
"If you had said to me a couple of weeks ago that we would take seven points in the first two games, after the bonus point win against Connacht last week, that would not have been too bad."
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