The 50-over role had become vacant following the retirement of Aaron Finch from the format last month.
Fast bowler Cummins, who has 73 ODI caps, has been the Test captain since last year and oversaw the Ashes victory over England in the last Ashes series at home.
Ben Oliver of Cricket Australia said: "We are very fortunate to have a number of high-quality leaders and senior players across all formats.
"The board and selectors agree Pat is the ideal choice to lead the ODI team through the next period, including the 2023 World Cup."
Cummins said: "I have thoroughly enjoyed playing under Finchy and have learnt an enormous amount from his leadership.
"They are significant shoes to fill although we are extremely fortunate to have a one-day squad with a huge amount of experience."
Australia's chairman of selectors, George Bailey added: "Pat has done an excellent job since taking on the captaincy of the Test side and we look forward to him leading the one-day team to the 2023 World Cup in India."