Ken Owens, DTH van der Merwe and Gareth Davies all crossed for the Scarlets but it was Alex Cuthbert's late try that sealed a 29-27 win for the Blues.
In a crucial second-half period the Scarlets had both Steffan Evans and Emyr Phillips sent to the sin bin while young lock Lewis Rawlins also saw yellow in the first half.
And Pivac - whose side did at least remain top of the table thanks to the losing bonus point although Leinster in second do have a game in hand - insists that their ill discipline must be resolved .
"Firstly, we're disappointed not to have won the match," he said.
"There were stages where we were taking a bit of control but discipline let us down at times and allowed the Blues back in and ultimately to win the game.
"We're disappointed with lapses of indiscipline from some of our players. We have to be better than we are in terms of our discipline, the boys know that."
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