Wooods has previously served as a vice captain on the 2016 Ryder Cup team and 2017 Presidents Cup team.
Els and Woods matched par on the first extra hole.
The 2019 Presidents Cup will be the 13th match in the competition's history.
"Ernie carries the flag for worldwide golf as a legendary figure both on and off the golf course, while Tiger has been one of the most transcendent athletes the sport has ever seen".
Woods takes over from Steve Stricker who led the 2017 USA Presidents Cup team to an emphatic win late a year ago in New Jersey.
The Presidents Cup is a biennial contest that has been played since 1994, with Team USA winning ten of the previous 12 contents. "This is something I've always wanted to do, I've always wanted to be a captain, and I'm very fortunate to have been given this opportunity".
"This is a huge honour for me", Woods said on Tuesday ahead of the Arnold Palmer Invitational. The core group of players on the U.S. Team have talent a lot of talent, enthusiasm and great team camaraderie.
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Tiger Woods and Ernie Els will duel in the Presidents Cup again, this time as captains. The International team's only victory was in 1998 at Royal Melbourne.
"I keep getting just a little bit better and a little bit sharper", Woods said, per CNN.
Woods has been named by Jim Furyk as a 2018 USA Ryder Cup vice-captain along with Stricker and Davis Love 111.
"I've had just the most incredible times and been able to play around the world.to do this with Ernie is a huge honour". That's more than any other player.
Woods meanwhile said his appetite to captain the U.S. team had been fueled by a stint as Steve Stricker's assistant in 2017. One person with knowledge of the decision said Woods reached out to the PGA Tour about his interest in being the next captain.
Els, a four-time major victor, takes over from Nick Price who has guided the Internationals at the last three Presidents Cups, including previous year when the United States romped to a 19-11 victory on home soil at Liberty National in New Jersey.
They also have battled plenty in tournaments.
In addition to the announcement of the two captains, officials also unveiled minor changes to format requirements and eligibility for the 2019 Presidents Cup. This is a change from past years where players were required to play twice during the same timeframe. Captains will later select four captains' picks (at date to be determined) closer to the December 2019 event.