Jennings, 24, replaces Mark Stoneman in the role, who left the county to join Surrey for the forthcoming season.
All-rounder Paul Coughlin will be vice-captain, while Paul Collingwood remains first-class and T20 Blast skipper.
"It's one of those dreams as a young boy, you want to lead your side," Jennings said. "It's an honour to lead Durham this year."
He added: "The couple of games I did last year were really good fun and it was a privilege and honour to do it."
Durham's opening One-Day Cup match is against Derbyshire at Chester-le-Street on 27 April.
Jennings' rise with Durham saw him win both club player awards last season, the Cricket Writers' player of the year award and ended 2016 making his England Test debut against India.
The South Africa-born batsman will also captain England Lions on next month's tour of Sri Lanka.
"Keaton has developed immensely over the last 12 months, not only as a player but also as a leader in the dressing room, and he has commanded the respect of the dressing room," Durham head coach Jon Lewis said.
"This new role reflects his standing in the side and we are confident he can lead us forward in the 50-over competition."
Lewis' side begin their 2017 season in Division Two of the County Championship with a 48-point deduction following financial issues.
They will also start the T20 Blast competition with a four-point penalty and the One-Day Cup with a two-point deduction.
Despite a reduced salary cap the county has signed two overseas players to cover the whole season, with South Africa Test opener Stephen Cook playing the first part of the campaign.
He will then be replaced by New Zealand batsman Tom Latham when Cook is expected to join up with South Africa for Test series against England which begins in July.
"I'm definitely looking forward to it," Jennings added. "The two signings we have made, Lathers and Cooky are brilliant for the squad and the guys as well.
"There is a real sense of positivity around at the minute. Hopefully we can push forward and see how far we can go."
Durham have named England batsman Keaton Jennings as captain for the One-Day Cup competition in 2017.