Contepomi, who worked alongside Cheika for 18 months, has been contracted through to the 2027 Rugby World Cup in Australia.
Cheika guided Argentina to the semi-finals of the recent Rugby World Cup in France and was an assistant and adviser to the team for two years before taking charge in March 2022.
Under the Sydneysider, the Pumas beat the All Blacks in New Zealand, the Wallabies in Australia, and England at Twickenham.
They knocked out Wales in the World Cup quarter-finals before losing to the All Blacks in the semis.
"Having been the head coach of Los Pumas fills me with pride, and it is one of the experiences that I enjoyed the most in my career as a coach," Cheika told the Argentina Rugby Union (UAR) website.
"Although I was born in Australia, a big part of me will be Argentina. I am convinced that Felipe and his staff are going to lead the team in the best way."
Michael Cheika (centre) coached the Wallabies for five years and took them to the 2015 RWC final. (Dave Hunt/AAP PHOTOS)
Cheika, who led the Wallabies for five years, coached Australia to the final of the 2015 World Cup.
Contepomi was the five-eighth at Irish club Leinster - and Cheika the coach - when they won their first European Cup in 2009.
He played 87 tests, and was key to Argentina reaching their first Rugby World Cup semi-finals in 2007. Four years later, he captained the Pumas at his fourth World Cup.
Contepomi paid tribute to Cheika, calling it an honour to have worked under the Australian as Argentina's attack coach and being "grateful for the extraordinary opportunity".
After he retired from playing in 2014, he learned the coaching ropes with the Argentina XV and Super Rugby Jaguares, before returning to Leinster. Cheika brought him home in 2022.
UAR president Gabriel Travaglini thanked Cheika.
"His professionalism, passion and dedication to Argentine rugby is something we will remember forever," Travaglini said.
"Today, we are convinced that Felipe is the right person for this new path towards Australia 2027."
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