Mooar has been at the Super Rugby champions for three years and helped the Crusaders to a second straight Super Rugby title in 2018 alongside head coach Scott Robertson.
He was previously head coach of the Southland Stags as well as backs coach with South African sides Eastern Province Kings and Southern Kings in the Super 15.
The 44-year-old Mooar will join Scarlets in 2019, with Pivac taking over from Gatland as Wales coach at the end of next year's World Cup in Japan.
"The opportunity to take on the role of head coach at the Scarlets is simply too good to pass up and is both an exciting challenge for myself, and a fantastic opportunity for my family to experience life on the other side of the world," said Mooar.
"The Scarlets is a rugby brand that's known the world over, it is a winning club with an extremely proud history, passionate fans, and is aspirational in its goals.
"It is humbling to be offered the opportunity to take the reins after the great work Wayne Pivac has led, as he moves on to the Wales job."
Pivac has been in charge at Scarlets since 2014 and helped the region win the Pro12 title in May 2017.
Scarlets general manager Jon Daniels said: "Brad's rugby philosophy, leadership style and personality are everything we were looking for in a head coach.
"Brad has a great understanding of what is required for success and how to build a winning culture.
"Throughout the process we have been hugely impressed with Brad's appreciation of Scarlets' place in the community, his passion for developing players and the research he had done on our heritage."