Gatland, who has also doubled up as the British and Irish Loans coach for the past two campaigns, has long intended to step aside after the tournament in Japan following more than a decade in the job.
The Welsh Rugby Union has been seeking a suitable replacement, and the search is now over as Pivac - promoted into his current role 22 months ago - will start working for the governing body from July 2019.
"We have secured the best man for the job and we have done so rigorously and decisively," Phillips said at a press conference. "There is no underestimating the positive benefit to be gained from having the time to plan properly for the future.
"We have avoided the feeding frenzy that can come at the end of a RWC year and we have been meticulous in ensuring we have someone of the talent, experience, charisma and rugby acumen to do the very best possible job for Welsh rugby. The handover process is something we will plan carefully and commence in detail next summer."
Pivac steered Scarlets to the Pro12 title with a 46-22 victory over Munster in Dublin two months ago, securing the Welsh region their first major trophy in 12 years.