Pogba was off target with a ninth-minute curler while Rashford was similarly wide from an ambitious effort with 17 minutes on the clock.
In the 24th minute, United earned a free-kick on the edge of the Derby box but Mata's effort was saved brilliantly by Carson.
But Jose Mourinho's side never gave up, with Lingard finally finding the back of the net in the 84th minute after a superb strike from outside the box into the top corner of the net. Sam Winnall then dragged one wide of the United post just before the half-hour mark as the visitors started to enjoy themselves.
On the other hand, the Rams are now unbeaten in their last seven matches and are second in the Championship with a fine shot of getting a chance to coming back to the Premier League.
Brilliant Kyrgios too good for Dimitrov in Brisbane
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. Scott Carson played well and the post played well for them.
It caused no surprise in the stands that Henrikh Mkhitaryan was brought off at the break for Romelu Lukaku, who missed his Everton homecoming on New Years Day after a nasty head injury on his last Old Trafford outing against Southampton.More news: Trump accuses ex-Clinton aide of "disregarding basic security protocols"