Gabriel Barbosa was felled in the box after half an hour and put the home side ahead from the penalty spot.
Yet Cirino levelled in the 63rd minute and put the visitors ahead also from the spot in the 71st.
Athletico looked on course for victory until Bruno Henrique scored in the 89th minute before Rodrigo Caio sent the 50,000 home fans into raptures with a stoppage-time winner.
The result leaves Flamengo fifth in the table on 10 points, three clear of Athletico in 10th after six matches.
Current champions Palmeiras are top of the league with 16 points.
(Reporting by Andrew Downie; Editing by Toby Davis)
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