However, having lost their last six games in a row, Derby staged a remarkable fightback in the second half to get enough, which proved enough after Cardiff drew against Rotherham.
United remain in sixth in the table but are now 11 points behind fourth-placed Aston Villa andsix behind Tottenham, in what could turn out to be a fifth Champions League spot, having played a game more.
Ivory Coast came from behind to beat Nigeria, thanks to second half goals from Franck Kessie and Sebastien Haller. William Troost-Ekong did score for the Super Eagles in the first half.
"I had to choose between him [Matic] and Pogba and I decided to start with Nemanja, as a holding six he's one of the best players we have in possession of the ball so that's why I went with him," Rangnick told MUTV.
City continued to pour forward in search of a winner, but were undone by a slick counter-attack when Dallas latched onto Ezgjan Alioski’s through ball and nutmegged Ederson to seal a smash and grab victory in the 91st minute.
A rare feeling for Chelsea, who are yet to record a loss in four matches now. Goals from Oscar, Willian and Costa secured a 3-0 win over Crystal Palace. It will be Chelsea's first win under Hiddink and their first of the new year.
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Divock Origi and Mamadou Sakho headed in two goals in as many minutes at the end of the first half. Everton lost Ramiro Funes Mori for a nasty challenge on Origi, who had to be replaced by Daniel Sturridge.
"What is for me unbelievable in the first half is that Man United has defended so badly on the flanks," Wenger told beIN Sports. "From the first minute on Diogo Dalot got a yellow card and Sancho and Antony did not contribute defensively.
"In the first half they have the goal and no more. In the second half, they have one attack with Harry Kane at the start and they score. "We try to score but we conceded the goal in extra-time. It's difficult when they defend so deep.
Isco and Sergio Ramos scored before half time, to turn the game on its head. Late goals from Lucas Vazquez and Luka Jovic sealed the points for Madrid, who moved six points clear of rivals Barcelona who are in second place.