Jaidon Anthony put Leeds in front with an imperious finish just after the half-hour mark following a brilliant solo run.
However, momentum switched in the second half before Randell equalised after a fine counter-attacking move.
Daniel Farke’s side had several chances to find a late winner, the best falling to substitute Patrick Bamford, but they could not replicate their 2-1 Championship victory in November.
Plymouth have not reached the FA Cup fifth round since the 2006-07 campaign and will get a second chance to make it into the last 16 with a replay at Home Park.
After a slow start for Leeds, Anthony sprung the hosts into life, breaking free of the Argyle backline before unleashing a strike that smashed off the post.
The woodwork was rattled again seconds later as the ball fell straight into the path of Wilfried Gnonto, whose fine effort was tipped onto the crossbar by Conor Hazard.
Anthony proved a nuisance all afternoon for the Pilgrims’ defence, and he opened the scoring with a superb bending strike into the right-hand corner after a fantastic solo run.
The goal proved a fitting tribute to the Bournemouth loanee’s late mother who passed away last week, and the forward revealed an undershirt that read “rest in perfect peace mum” after finding the back of the net.
Plymouth came back at the Whites in the second half, with Ryan Hardie seeing a looping effort cleared off the line by Sam Byram.
Nevertheless, they did find their equaliser through Randell with 17 minutes to go.
Anthony tried to replicate his first-half heroics with another bending effort before Bamford saw his close-range effort saved by Hazard.
While Leeds could not convert their late chances, they maintained their unbeaten home record this season.
