Costa, whose late header earned Atletico a 1-0 first-leg win at San Siro, acrobatically put the Spanish side ahead with the first chance of Tuesday's last-16 second leg but former Real Madrid star Kaka headed Milan level.
Arda Turan's deflected strike made it 2-1 to Atletico before the break, and then second-half goals from Raul Garcia and Costa sealed a deserved 5-1 aggregate victory for the Spanish side.
Atletico coach Diego Simeone said: “We are happy with the work the lads did over the last two and a half years. We play as a team and I think that is our greatest strength, knowing our limitations and our strong points.
“It was dangerous for Milan to go on the counter-attack, so we had to hold on to the ball and beware of losing possession.
“In the first leg Milan played well and we had left them too many spaces to play in, which is what they enjoy.”
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