Glasgow, who have no fewer than 20 players on World Cup duty, did hit back through Michael Cusack and Glenn Bryce but Scarlets - despite ending the game with 11 men on the field due to a combination of yellow cards and injuries - held on for a 16-10 win.
"It was a very nervous time at the end there," Pivac said. "When you lose one player in the modern game it's pretty difficult to defend. Then when you go down to 13 men it is really difficult. It obviously gets increasingly difficult when down to 11!
"They just kept coming, it was unbelievable.
"I'm certainly very proud of the effort. We were down in numbers; we played particularly good in the first half into the wind and got in the lead.
"We always knew that Glasgow would come back and our defence would play a part. That's certainly the way it turned out."
Scarlets are seeking for a return to the top four for the first time since 2012/13. yet Pivac is not about to let his side get carried away after their opening victory.
"Every team wants to be in the play-offs and we're no different," added Pivac. "We're not looking beyond the World Cup period at this stage. The squad has got a job to do and we'd like to be in the top four by the time the internationals return to us."
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