The Portugal forward grabbed a hat-trick in a 6-1 demolition of Galatasaray on the opening night and needed little persuading to add two more goals to his tally at the Santiago Bernabeu.
He gave the Primera Division club an early lead with a firm header but, after he doubled the advantage from a spellbinding Di Maria cross midway through the second half, his Argentinian team-mate arguably stole the show with a classy brace of his own to settle the tie.
Next up for Carlo Ancelotti's men is a double-header with Juventus, who rallied to draw 2-2 with Galatasaray, while Stale Solbakken's Copenhagen take on Gala looking to build on the single point they earned against the Italians two weeks ago.
There was no Gareth Bale in the line-up after the Welshman was ruled out with a thigh injury, but Real's original wing wizards managed just fine on their own.
Marcelo was also a constant menace to the Danish side and inside six minutes he had fed Karim Benzema for a header that just darted wide.
Di Maria then picked out Ronaldo for a speculative effort before Nicolai Jorgensen blocked Marcelo's left-footed drive.
Copenhagen could only resist for 21 minutes, though, with Ronaldo plundering the lead on his 100th appearance in European club competition.
Marcelo provided the cross for the Portuguese, who stooped to nod the ball beyond Johan Wiland.
The visitors had been firmly under the cosh but did enjoy two decent chances in the first half.
Yussef Toutouh managed to sneak through the Real backline to warm Iker Casillas' hands on the half-hour mark and Jorgensen clattered the bar with a header five minutes before the break.
But the hosts reasserted their authority following the restart, Di Maria inspiring a smart stop from Wiland before Dani Carvajal screwed a right-footed effort just over.
When the hour mark ticked past, Wiland thought he had done enough to preserve the narrow deficit when he smothered Pepe's header from Di Maria's free-kick.
But it was a false dawn as Ronaldo arrived in the six-yard box to head home another goal.
This time the Copenhagen defence were caught cold when Di Maria managed to fire the ball across to his team-mate with his left boot in a spectacular display of improvised skill.
Perhaps they had been struck dumb by Di Maria's wizardry, or maybe it was just a case of broken resolve, but the stricken Danish backline soon conceded a killer third.
Provider turned goalscorer in the 71st minute as Di Maria stole onto Karim Benzema's pull-back from the left to dance into the D. He then left Wiland sprawling with a sublime finish into the far corner.
Madrid might normally have eased up at that point but their tails were up in front of a bouncing home crowd.
Di Maria was especially eager to match his Portuguese colleague's achievement and did so one minute into stoppage time, switching his feet to guide the ball across the hapless Wiland with his favoured left boot from the edge of the area.