The draw, made tonight by former England defenders Micah Richards and Tony Adams at the Etihad Stadium, pits Marcelo Bielsa's men against the currently managerless Gunners, away from home, over the weekend of 3 to 6 January 2020.
"I'm sure they'll go there with confidence, especially with Bielsa," said Richards.
United currently boast an Arsenal striker in their squad, loan signing Eddie Nketiah. The frontman will have to sit the game out under the terms of his loan deal.
The two sides last met in the third round in 2012, when Arsenal were 1-0 winners.
Leeds were 1-0 winners at Highbury in the competition in 1997.
All 32 of this season's FA Cup third round ties will kick off a minute late as part of a campaign to promote mental health.
In an attempt to encourage football supporters to think about their mental health, the Football Association has partnered with the Heads Up campaign and the National Health Service's Every Mind Matters.
Marcelo Bielsa's first taste of the FA Cup last season brought a 2-1 third round defeat at Queens Park Rangers.
United won the FA Cup in the 1971/72 season and have finished runners-up on three occasions.