Atalanta director general Umberto Marino was unhappy with what he believed was a "serious error" to allow Nikola Milenkovic's 93rd-minute winner to stand as Fiorentina triumphed 3-2 at Gewiss Stadium in the Coppa Italia quarter-finals.
A ninth-minute Krzysztof Piatek penalty was overturned by goals from Davide Zappacosta and Jeremie Boga to give Atalanta the lead.
Piatek equalised with 20 minutes remaining after having a penalty saved by Juan Musso, before Milenkovic hit a superb volleyed winner in the last few seconds of stoppage time.
La Viola – who also had Lucas Martinez Quarta sent off with 10 minutes remaining for a second yellow card – celebrated wildly, but the hosts were unhappy the VAR chose not to overturn it, believing Giacomo Bonaventura was offside in the build-up.
Speaking after the game, Marino told Sport Mediaset: "I can only congratulate Fiorentina and accept what happened. There are games when you get the feeling it is an uphill struggle for you and nothing is going to go your way.
"Fiorentina always seem to have penalties against us, I don't know if they were the right decisions or not, but it's the third goal where we have serious problems.
"Merih Demiral told me there was no foul for the free-kick that led to the goal, but I also have to ask if that Marten de Roon touch was sufficient to put Bonaventura back onside, then I have evidently understood nothing about football over all these years.
"Mistakes can happen even with VAR, unfortunately they are happening a bit too often, including in Serie A. Not seeing an offside like that is in my view a serious error.
"It is disappointing to miss out on a semi-final because of an incident like that in the 93rd minute. It is a serious error, I repeat, but errors are part of sport.
"We simply have to accept them and move on."
Fiorentina boss Vincenzo Italiano admitted he took a risk on the set-piece that led to the winner, sending several players forward in an attempt to avoid extra time with a man fewer.
"It was a game where I think before their equaliser, we were controlling the situation and had chances to go 2-0 up, but they equalised at the first opportunity," he said.
"At the end, I called everyone forward on the free-kick because there was genuine fear of playing extra time down to 10 men. We were aggressive, brave and lucky too to believe right to the very end."