But with just a minute left Iain Henderson went over for the home side's second try - adding to an early touch down for Louis Ludik and three Paddy Jackson penalties - to lead Ulster to a 19-8 victory.
The result lifted Ulster to the top of the table while Scarlets have started their campaign with three defeats, with Pivac urging his troops to start taking their chances.
"It was a step up, I thought, effort wise we couldn't fault. When we got down to 14 we were able to hold them out so a lot of good work was done there," said Pivac.
"Ulster are a very good side, we knew that. We've had some real ding-dong battles over the last couple of years with them and that one out there was no different, it's just frustrating to repeat the fact that we're getting numerical advantages but we're not nailing sides.
"We're not clinical enough in our finishing. We are creating opportunities but that's the third week in a row we haven't put sides to bed with those opportunities. To come away from the game with nothing is a bitter pill.
"The boys know that they're creating scoring opportunities but they're not finishing them off, that's the frustration. But if we weren't creating opportunities it would be a lot worse. We've got to take the positives out of that.
"We've got Connacht followed by Treviso and then the Dragons, I think it's a simple equation for us - it's got to be three from three.
"We can't fault the effort, the boys really threw it in there to come back from 11-0 and could have potentially turned the game around."
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Guinness PRO12 Scarlets News
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