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Scott Laird's deflected 47th-minute strike gave third-tier Preston the scent of a huge upset but United regrouped and goals by Ander Herrera, Marouane Fellaini and Wayne Rooney sent the Premier League side through.
"We showed an unbelievable team spirit," United manager Louis Van Gaal told the BBC.
"We changed the shape and that was the solution. We scored fantastic goals, a little bit lucky also I think. I liked the second goal a lot, we have trained for that a lot.
"Marouane Fellaini has a body, he also scores goals. I played him in the position that he wants for the first time but I do not always have a position for him. The spirit was better."
Preston manager Simon Grayson led Leeds United to a famous victory over Manchester United in an FA Cup third-round tie at Old Trafford in 2010 and his organised team worked tirelessly to give themselves a chance to emulate that feat.
They restricted United with disciplined defending and the visitors failed to trouble keeper Thorsten Stuckmann in the first half.
Laird's speculative strike, which burst through the grasp of United goalkeeper David de Gea, sent the home fans into a frenzy and Van Gaal's last realistic chance of winning silverware in his first season at United looked in jeopardy.
But the introduction of the livewire Ashley Young in place of ineffective striker Radamel Falcao revived the visitors and good work by the winger down the right set up the chance for Herrera to squeeze in a 65th-minute equaliser off the post.
Antonio Valencia's right-wing cross was powerfully met by Fellaini at the far post and the towering Belgian lashed home the rebound seven minutes later.
Rooney chased a through ball and narrowly won the race with Stuckmann whose clumsy tackle led to a penalty which the striker gleefully lashed home to register his first goal of the year and stay on course for his first FA Cup winners' medal.