Leeds will face the winners of a replay between the sides after National League club Eastleigh held Bolton to a 1-1 draw at the Silverlake Stadium on Saturday.
Eastleigh were on the verge of a huge upset and led 1-0 on a rain-ravaged pitch until Darren Pratley rescued Bolton with an 87th-minute equaliser.
United, who came safely through a third-round clash with Rotherham United by winning 2-0 at Elland Road, will learn their opponents when Eastleigh travel to the Reebok Stadium for a replay next Tuesday.
Leeds’ last meeting with non-league opposition in the FA Cup ended in a grim defeat to Histon during Gary McAllister’s reign as manager in December 2008.
Bolton, however, are in the middle of a grave financial crisis owing to debts of more than £150m.
The Championship club face a winding-up petition and have failed to keep up payment of wages to their players. They are attempting to cash in on young striker Zach Clough.
Speaking after his goal at Eastleigh, Pratley said: “We need the money as much as Eastleigh. As much as it’s nice for Eastleigh to get the money, it’s nice for us as well.
“A draw will probably get both clubs some money, maybe a televised game, and it can keep us afloat for a bit. It’s important financially. We need to stay in the hat.”
United boss Steve Evans had been looking for a home draw with a top Premier League side in round four.
Following Leeds’ win over Rotherham United, Evans said: “Every coach would want the lowest ranked team at home or one or the big giants.
“It’s one of those days I’m yet to experience, where there’s 35,000 in Elland Road. The place has been rocking with less but that would be the preferred choice for me.”
The meeting with Eastleigh or Bolton will take place on the weekend of January 30.