Diego Simeone's men, who won the first leg 1-0, triumphed at the Vicente Calderon as goals from Costa and Arda Turan sandwiched Kaka's close-range header in the first half.
Raul Garcia and the Spain striker's second added further gloss to the scoreline inside the final 20 minutes, with the home side good value for the win.
It took the hosts just three minutes to double their first-leg advantage. Koke curled a sublime cross towards the far post and Costa showed great athleticism to divert a side-footed volley past Christian Abbiati.
Milan settled into the encounter after a lackluster start and was back in the tie on 27 minutes with the equalizer on the night.
The ball was spread wide to Andrea Poli and he stood up a pinpoint cross to the back post where Kaka headed home from close range.
The Brazilian then missed a clear chance to put Milan ahead on away goals, as he leapt to meet Adel Taarabt's center, but succeeded only in heading over from six yards.
That miss proved costly as Turan restored Atletico's lead in fortuitous circumstances five minutes before the break. The Turkey international's volley took a huge deflection off Daniele Bonera before looping over Abbiati and nestling in the bottom corner.
Raul Garcia nearly made it three on the stroke of halftime with an acrobatic free kick that went just wide.
Atletico started the second half in the same swift fashion that it had started the first and captain Gabi smashed a 20-yard effort against the crossbar on 48 minutes.
Shortly after the hour mark, Costa held up a ball from the byline and Miranda headed straight at Abbiati from eight yards.
The pressure finally told when the tie was put beyond all doubt 20 minutes from time, as Raul Garcia rose high to nod Gabi's free kick into the bottom corner.
Robinho went close to reducing the deficit when he cracked an effort against the crossbar, but there was still time for one last piece of action as Costa fired across goal and into the bottom corner in the 85th minute to seal a resounding win for Atletico.