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.
Scott Carson's first terrific free-kick stops came on 21 minutes after Juan Mata was floored by Alex Pearce whilst trying to complete a one-two 20 yards out, and the subsequent set-piece was tipped wide for a corner before it could find the top left corner.
Lingard scored the first goal of the game in the 84th minute with a superb half volley from the edge of the Derby box before Lukaku sealed the encounter with a close range finish after playing a clever one two with Anthony Martial. But Sam Winnall failed to profit as he dragged his shot wide.
Rashford's finishing let the hosts' approach play down, though, when the striker skied over the bar after being played through.
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Manchester United had the first attempt on goal through captain Pogba firing a straight shot at Carson in the second minute.
United also got good news regarding Romelu Lukaku, as the Belgian global will be fit and available to play after suffering a scary blow to the head against Southampton on December 30.
Scott Carson pulled off another save to deny Jesse Lingard from the edge of the area before Paul Pogba trickled an bad effort wide on the rebound.
The Red Devils hit the woodwork again minutes later from Romelu Lukaku's header, but a goal wouldn't have counted anyway as the big Belgian was ruled offside.
The barrage continued and Pogba wastefully side-footed wide with the goal gaping when following up Lingard's saved effort, before Derby finally yielded with six minutes to play.