The winger struck late on to give Diego Simeone's men a big early win in their bid to defend La Liga and again upset their city rivals
The reigning champions started brightly at the Santiago Bernabeu in the seventh meeting between the sides since the turn of the year and made a breakthrough in the 10th minute when Tiago headed home from a Koke corner.
Real battled their way back into the clash and had the opportunity to pull themselves level when Cristiano Ronaldo was felled by Guilherme Siqueira in the penalty area.
Ronaldo – making his return from injury – coolly stroked past Miguel Angel Moya from 12 yards after 26 minutes.
Karim Benzema squandered numerous opportunities to edge Real ahead before Carlo Ancelotti's men were punished in the 76th minute when substitute Arda scored the winner.
The result means Real have now lost two of their opening three La Liga fixtures, while Atletico – watched from the stands by coach Diego Simeone as he serves a touchline ban - move to within two points of early leaders Barcelona.
Ancelotti described Ronaldo as being in "optimal condition" this week following a hamstring problem that kept him out of Portugal's opening Euro 2016 qualifier, and the attacker slotted straight back into the Real starting XI.
However, it was the visitors who looked the more threatening in the early stages, with Koke sending a couple of dangerous crosses into the Real penalty area.
And it was the 22-year-old who provided the delivery when Atleti broke the deadlock in the 10th minute – Tiago evading his marker to head in at the near post from a corner.
James Rodriguez is yet to score a league goal for Real since making his big-money move from Monaco during the close-season, and the Colombian squandered an opportunity to open his account five minutes later.
From a Ronaldo cross, Rodriguez volleyed the ball at the turf before watching as it looped harmlessly over the crossbar.
Gareth Bale drew a superb diving save from Moya in the 24th minute with a curling free-kick from 20 yards, but the Atleti goalkeeper was left to pick the ball out of his net moments later.
Siqueira brought down Ronaldo in the penalty area and the Portugal star made no mistake in sending Moya the wrong way as he rolled the spot-kick into the bottom left-hand corner.
Benzema's heavy touch thwarted an incisive Real counter-attack in the 34th minute, and the Frenchman endured a frustrating first half – dragging wide of the left-hand post and seeing a header kept out by Moya before the interval.
Javier Hernandez – on loan at Real from Manchester United – came off the bench to make his debut just after the hour mark before Atleti came close to re-taking the lead when substitute Arda fizzed an effort agonisingly wide of the right-hand upright.
The Turkey international made full use of his second chance, though, sweeping a low strike into the bottom-left corner after a pull-back from Raul Garcia to seal all three points for Atletico.