Gian Piero Gasperini's men travelled to Valencia on the back of a comprehensive 4-1 win at San Siro and that coupled with the efforts of the exceptional Ilicic at Mestalla took Atalanta through to the last eight, the two sides playing out a thriller despite no fans being in attendance as a consequence of the coronavirus outbreak.
Atalanta went into the break with a deserved lead, as Ilicic won and scored a pair of penalties either side of a poacher's effort from Kevin Gameiro.
Gameiro got a second to pull Valencia level and Ferran Torres put them in front for the first time on the night, but Ilicic completed his hat-trick with a fine strike before earning the visitors the victory with another excellent effort.
Ilicic showed a sign of things to come after just three minutes as he lured Mouctar Diakhaby into a clumsy challenge with a clever turn after a brilliant solo run, and the Slovenia international converted the spot-kick confidently.
Gameiro got Valencia back on level terms for the match, slotting into an empty net despite Jose Luis Palomino initially doing well to cut out Rodrigo Moreno's throughball.
But Atalanta were in front again just before half-time – Ilicic dispatching another penalty after his run into the box was halted by Diakhaby handling the ball.
The second half began at breathtaking speed and Valencia were fortunate to not concede again, as Remo Freuler's long-range effort struck the crossbar, but the hosts equalised less than a minute later – Gameiro heading in Ferran's delivery.
Ferran then got himself on the scoresheet with a clever finish over the approaching Marco Sportiello, only for Ilicic to deny Valencia a victory.
Ilicic finished clinically after beating Dani Parejo on the edge of the box, before completing the scoring by caressing a first-time shot into the top-left corner with eight minutes remaining.