The move is seen as part of Ukraine's efforts to reinforce its army as soldiers struggle to hold positions against Russia
Spaniard Lirola collected Erik Pulgar's long drilled pass before charging towards goal and slotting home his first Fiorentina goal six minutes from time, winning the tie for the hosts despite German Pezzella's sending off 14 minutes earlier.
Patrick Cutrone also scored his first goal for the club when he put Fiorentina ahead in the 11th minute of his first start since arriving on loan from Wolverhampton Wanderers last week.
The Italian laid the ball off to Dalbert, who pulled back a perfect pass that allowed the forward to stroke the ball into an empty net.
Atalanta should have been level when Mario Pasalic crashed Ruslan Malinovskiy's cross against the bar from point blank range.
The away side were much improved after the interval and went close twice around the hour mark. Pasalic was denied by the onrushing Pietro Terracciano before Robin Gosens hit Luis Muriel's cross against the post with the goal at his mercy.
They scored their deserved leveller when former Fiorentina man Ilicic turned in another superb Malinovskiy cross.
Things grew tougher for the hosts when Pezzella was sent off after being caught diving and receiving a second yellow card, but Lirola won the tie to set up a tough match with Serie A title chasers Inter.
Inter thumped Cagliari 4-1 on Tuesday. Romelu Lukaku opened the scoring after just 21 seconds and headed home another early in the second half to book Antonio Conte's side a spot in the quarters.
Further goals from Borja Valero and Andrea Ranocchia helped Inter bounce back after a 1-1 home draw with Atalanta at the weekend that let Juventus take top spot in Serie A.
The winner of the last eight clash later this month will have a semi-final against either Lazio or Napoli who comfortably saw off second-tier sides Cremonese and Perugia on Tuesday.
Holders Lazio, who are on a club-record run of 10 straight wins in Serie A, made short work of Cremonese in their 4-0 victory in Rome.
Troubled Napoli eased into the quarters with a 2-0 win over Perugia thanks to two Lorenzo Insigne penalties.
Juventus host Udinese and AC Milan take on SPAL later on Wednesday, with Roma travelling to Parma on Thursday.
Torino are already in the last eight after beating Genoa on penalties last week.
The move is seen as part of Ukraine's efforts to reinforce its army as soldiers struggle to hold positions against Russia
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