Contepomi, who won 87 caps as a player for Argentina, served as assistant to Cheika, who leaves after less than two years in the role having led them to the semi-finals of the World Cup in October and a fourth-placed finish.
"I want to thank all the Argentines for these wonderful years together, the UAR, the staff and the players for having trusted me for this challenge," Cheika said in a statement.
"I am very proud to have been the head coach of Los Pumas and it is one of the experiences I have enjoyed the most in my coaching career.
"Although I was born in Australia, a big part of me will be Argentina. I am convinced that Felipe and his staff will lead the team in the best way."
Cheika, who was appointed in March 2022, led Los Pumas in 24 test matches, recording 11 wins and 13 defeats.
They reached the last four of the World Cup in France, where they suffered a heavy defeat to New Zealand before narrowly losing to England in the bronze final.
"I had the honour of working with Michael Cheika in the consolidation of a sporting project, which challenges and excites us as part of a great Argentine team," Contepomi said.
"For that reason, I am grateful for the extraordinary opportunity that the UAR gives me to continue working towards the cherished goal of the 2027 World Cup in Australia."
