Woods will take charge of Team USA at Royal Melbourne Golf Club from December 9-15, while Els skippers the worldwide side at the venue where they claimed their only Presidents Cup victory in 1998.
Woods, who holds the matchplay competition's record with 24 match wins for the United States, and Els, who has won an global side best 20 matches, will lead their teams into the 2019 edition on December 12-15 at Royal Melbourne.
They are expected to be introduced Tuesday at the Arnold Palmer Invitational near Orlando, Florida.
"After working as a captain's assistant for Steve in 2017, I realised that I wanted to captain the 2019 event at Royal Melbourne", said Woods. The last time he and Els went head-to-head in Presidents Cup action was in 2003 in South Africa when the two were picked for a sudden-death playoff.
I've played in the Presidents Cup eight times and I had a chance to serve as vice captain a year ago.
"To be named captain of the team is an incredible honour".
"I had a good shot at winning the tournament", Woods said in a post-round interview on CBS.
"I've had just the most wonderful times and been able to play around the world.to do this with Ernie is a huge honour".
"No other city outside of the United States has hosted the Presidents Cup more than once - and we can't wait to welcome it back for an unprecedented third time next year", Minister for Tourism and Major Events John Eren said.
Woods was an assistant captain on last year's American team that won in blowout fashion.
Woods has finished in the top 15 in each of his last two PGA Tour starts.
In 2018, Melbourne will also host the 59 edition of the World Cup of Golf at The Metropolitan Golf Club from November 21-15. The U.S. won 19-11 at Liberty National in New Jersey previous year, boosting its overall record to 10-1-1. They spoke on condition of anonymity because the captain selections have not been announced. Additionally, the top eight players for the global and U.S. Team standings will officially qualify following the completion of the 2019 PGA TOUR Championship.