The breakthrough would eventually come on 84 minutes, with Lingard unleashing a cracking effort from outside the box after a lay-off from sub Romelu Lukaku.
United's best chance fell to Rashford two minutes later, but he fired over the crossbar after Juan Mata had forced Carson to save a free-kick headed for the top left corner of the net.
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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.
It looked set that Mourinho's men were going have have to travel to Pride Park for an unwanted replay with seven minutes to go, but the hottest man now on the block at Old Trafford had other ideas.
Lingard and Lukaku spared the home side's blushes, but questions will remain over United's ability to get over the line in games, with the 2-0 win over Everton last time out ending a run of four matches without a victory - including a shock 2-1 defeat to Bristol City in the EFL Cup quarter-finals. He played it to Martial on the right who put it back inside, and the no. 9 only had to slot it past Scott Carson inside the box to make it 2-0.
Lukaku played the ball into Lingard's path, and the attacking midfielder hit an unstoppable shot into the top-right corner.