Chiesa scored a stunning solo goal and set up the late equaliser for Luis Muriel after the visitors, who knocked out title holders Juventus in the previous round, had led 2-0 and 3-2.
It was less dramatic, however, than Fiorentina's Serie A match at home to Inter Milan on Sunday, when they recovered from 3-1 down to also draw 3-3 with a hotly-contested penalty converted in the 11th minute of stoppage time.
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Atalanta, who last won the competition in 1963, got off to a flying start thanks to two moments of inspiration from Josip Ilicic.
The Slovenian collected the ball in midfield, charged 60 metres and laid the ball off for Alejandro Gomez to score in the 16th minute before providing slide-rule pass for Mario Pasalic to add the second two minutes later.
Chiesa, whose father Enrico played in the Fiorentina side on the last occasion they won the competition in 2001, was not to be outdone, however.
In the 33rd minute, he won possession from Jose Palomino just inside his own half, burst forward and drilled the ball past Etrit Berisha.
Three minutes later, Marco Benassi levelled for the hosts when he volleyed home after Atalanta failed to clear a throw-in.
Yet Atalanta went back in front in the 58th minute when Marten de Roon rifled home the ball from the edge of the area after a Fiorentina clearance landed at his feet.
But Fiorentina levelled again in the 79th minute, Muriel turning in Chiesa's cross for his sixth goal in all competitions since joining Fiorentina on loan from Sevilla in January.
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