The hosts made a strong start as they took the game to the Premier League outfit and created the first real opportunity. Louis Sibley picked up the ball up in the United half and the 18-year-old drove forward before firing his effort just wide of the post.
Former United man Wayne Rooney then came close from a free-kick, he got his effort up and over the wall and it was heading towards the bottom corner but visiting keeper Sergio Romero showed excellent reflexes to claw the ball away.
United started to come into the game and Juan Mata and Ighalo both came close before they took the lead through Shaw. Bruno Fernandes and Jesse Lingard both saw shots blocked before the ball dropped to Shaw. The defender hit his first-time volley down into the ground and the ball looped up into the air, over stranded Derby goalkeeper Kelle Roos and into the net with 33 minutes on the clock.
Shaw turned provider for United’s second as half-time approached. He picked out Ighalo and it looked as if he had lost control between two Derby defenders but the Nigerian managed to poke a boot out and direct his strike into the bottom corner.
The Rams started brightly again after the interval and Max Bird came close before Rooney’s exquisite pass found Jayden Bogle down the right flank. Martyn Waghorn managed to get his head to Bogle’s cross, but his glancing effort drifted agonisingly wide of the upright.
Ighalo grabbed his second with 20 minutes of normal time remaining after being picked out inside the box by Mata. The January recruit’s initial effort was blocked, he latched onto the rebound and smashed home a left-footed finish into the roof of the net.
The Championship side refused to give in and Jack Marriott forced a fine save from Romero, while Rooney also saw another brilliant free kick that was destined for the top former blocked by the Argentine stopper in injury time but United held on to their clean sheet.
United will travel to Premier League strugglers Norwich City in the quarter-finals on the weekend of March 21.