Preston took a shock lead just after half-time through Scott Laird’s deflected effort from just inside the area, but United scored twice in eight minutes to claw their way back into the game.
Ander Herrera hooked home the equaliser on 65 minutes, despite Preston calls that an offside Wayne Rooney was interfering with play, before Marouane Fellaini struck from six yards to complete the turnaround.
Rooney added a third from the spot after being brought down by Thorsten Stuckmann, and Louis van Gaal’s side now have a home tie with Arsenal at Old Trafford in the quarter-finals.
Van Gaal started Rooney in a more advanced position than previous weeks, but United struggled to break down their League One opponents early on.
In a first half of few chances, Preston came closest through Joe Garner, whose header from a free-kick went over the bar after the midfielder got the better of Marcos Rojo.
Angel Di Maria, whose final ball rarely found a red shirt in the first half, did go close with a free-kick from 25 yards, but Preston worked hard to keep the Premier League side at bay.
Laird gave Preston a deserved 1-0 lead two minutes after the restart, seeing his deflected effort from just inside the area go through the hands of David De Gea, who should have done better.
Garner nearly added a second with an unmarked header from 10 yards, but the effort fell into the grateful hands of De Gea.
Radamel Falcao was poor on the night, forcing Van Gaal to replace him with Ashley Young on the hour mark, and the change made an impact.
Spanish midfielder Herrera equalised for United after being found by Young on the right, but Preston appealed for offside as his pulled effort crept in the far post past the offside Rooney.
The hosts felt the England striker was interfering with play despite not touching the ball, with goalkeeper Stuckmann delaying his dive until the ball went past Rooney.
Paul Gallagher curled a free-kick just over at the other end, but United went ahead through Fellaini on 72 minutes after being found by Valencia at the far post.
The Belgian looked to shove defender Tom Clarke in the back before seeing his header saved by Stuckmann, but referee Phil Dowd gave nothing as Fellaini slotted home the rebound.
Fellaini went close to a second minutes later, this time seeing his close-range effort saved by Stuckmann, before Rooney added a third from the spot with two minutes left.