The Welsh region looked a real threat with ball in hand in opening up a 24-15 half-time lead with tries from Harry Robinson, Rory Pitman and Gareth Davies.
Davies' second try after the interval meant that Pivac's troops were 14 points to the good but Ulster refused to give in and Andrew Trimble's last-gasp score, with Ian Humphreys' conversion, meant honours ended even.
And Pivac admitted his troops had only themselves to blame for not kicking off their season with a victory.
"Around 32-18 I think the game should have been won," he said.
"We've got to control our emotions on the field. And those are the things that we'll be working on pulling this tape apart as a team, to ensure we learn from that performance and are better next week
"Ulster are a good side, but it was more the gifting of points that was frustrating for me, rather than they created some great tries.
"On the other hand I thought we did create some good tries and I think we'll continue to do that. That's the type of team we are."
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