The Portuguese forward scored the opening two goals for the Blancos before his Argentine teammate added a third and fourth to the scoreline.
Copenhagen has endured a disastrous start to its domestic season and never looked like picking up only its second-ever away win in the Champions League as Ronaldo opened the scoring after 21 minutes at the Bernabeu.
The Portugal star — who scored a hat trick at Galatasaray on matchday one — added another in the second half after a magnificent piece of skill from Di Maria.
And the Argentine winger rounded off the win with a double of his own in the final 20 minutes to put the Blancos in firm control of Group B with six points from two games.
The Spanish giants suffered a Liga derby defeat to Atletico Madrid for the first time since 1999 on Saturday and coach Carlo Ancelotti responded by making five changes. Fabio Coentrao, Sergio Ramos, Alvaro Arbeloa, Isco and Diego Lopez dropped to the bench as Marcelo, Dani Carvajal, Raphael Varane, Luka Modric and Iker Casillas started, with the latter making his 129th Champions League appearance.
Copenhagen boss Stale Solbakken made three alterations from their own weekend defeat to Brondby, with Ragnar Sigurdsson, Rurik Gislason and Youssef Toutouh replacing Georg Margreitter, Cristian Bolanos and Fanendo Adi.
The visitors did a good job of keeping Madrid at bay in the opening stages, but the hosts eventually made the breakthrough in the 21st minute. Marcelo’s deep cross evaded goalkeeper Johan Wiland, giving Ronaldo an easy headed finish at the back post for his first of the game.
Di Maria then wasted two great chances to double the lead and Copenhagen almost capitalized on the hosts’ profligacy.
Four minutes before the break Casillas failed to catch a corner, allowing Nicolai Jorgensen to head toward goal, but the ball hit the crossbar before dropping to Modric for the clearance.
Ronaldo went close to getting his second in the 64th minute when his curling free kick from the edge of the area landed on top of the net, but he was not to be denied a minute later.
Di Maria got to the byline to produce a cross with an outrageous piece of skill, and Ronaldo burst in front of his marker to head home again.
And with 19 minutes to play, Di Maria made amends for his first-half misses with a wonderful finish as he cut in from the right and hammered a shot past the helpless Wiland.
Real saw out the final stages with consumate ease and then scored a fourth for good measure.
Di Maria pounced on a loose ball and held off his marker before smashing a low effort into the bottom corner from just inside the area, rounding off an eye-catching display from Ancelotti's men.