Lingard, in the form of his life, pulled off a stunning first-time effort from outside the area to break the deadlock on 84 minutes.
"The Championship is coming in a direction where they are Premier League level, they just play in another competition".
Three minutes later, Pogba had a chance to send United ahead but his curling shot from just outside the box towards the top corner missed wide.
United got its first win in five matches on Monday by notching two second-half goals to get a 2-0 win at Everton.
Romelu Lukaku, who had replaced Mkhitaryan at the break, then had his goal-bound shot blocked as Pogba curled off target in a good spell of attacking intent from United.
Meanwhile, Rowett responded to Mourinho's changes by finally bringing Vydra on with 23 minutes remaining.
Lukaku was next to head wide on 55 minutes, before Thomas Lawrence forced a diving save from Romero.
However, with a replay suiting neither side both teams pushed forward for a victor and Lingard hit a great strike six minutes from time to put United ahead.
That opener flattened out the game as United looked comfortable in possession, but Romelu Lukaku struck in stoppage time to double the lead and seal progression to the Fourth Round of the FA Cup.