Aquilani struck his third goal before the hour mark to hand Fiorentina a 3-2 lead only for Genoa to equalise 13 minutes from time.
"This was my first hat-trick so it was special," Aquilani said to Sky Italia.
"It's very satisfying to get a hat-trick but it's a real shame about the result.
"We really wanted to win and instead we have dropped two points."
A victory would have seen the Viola cut the gap with third-place Napoli to one point.
Napoli, who hold the last Champions League qualifying spot, were held to a 1-1 draw by Chievo on Saturday.
"It's a missed opportunity," Aquilani said. "It was important to win and go within a point of Napoli, but in Serie A all the teams are very difficult to beat.
"Our aim is to catch Napoli and if possible, overtake them in the standings.
"We just have to take it one game at a time.
"I believe we are having a strong campaign and hopefully we will win our next game."
Fiorentina are fourth in Serie A and travel to Sardinia to face Cagliari on Saturday while Napoli visit Genoa.