After conceding a penalty on Sunday, which Alberto Gilardino converted for Genoa, Montella's team equalised thanks to Alberto Aquilai's own spot kick only to go behind again a minute later following Luca Antonini's strike.
While Aquilani made it 2-2 before half-time and then completed his hat-trick in the second half to put Fiorentina ahead in Florence, Genoa claimed a share of the points through Sebastian De Maio in the 78th minute.
Both Aquilani and Facundo Roncaglia had chances to add a fourth goal to Fiorentina's tally but they failed to convert and Montella was clearly bemused after the match.
"I think we could've won if we had made more of certain incidents," the 39-year-old coach told Sky Sport Italia.
"I saw Facundo Roncaglia's late shot and it curved in a bizarre fashion, almost as if it had been remote-controlled to swerve wide.
"It was incredible. Maybe we're cursed."
The draw meant Fiorentina failed to gain ground on third-placed Napoli, who had drawn 1-1 with Chievo on Saturday.
Fiorentina (41 points) remain fourth, three points behind Napoli in the chase for Serie A's third UEFA Champions League spot.
While Fiorentina have struggled with injuries in the past few months with the likes of Giuseppe Rossi, Mario Gomez and Borja Valero on the sidelines, Montella received a boost with Brazilian midfielder Anderson making his debut off the bench against Genoa.
The 25-year-old played the last 16 minutes after replacing Aquilani and Montella was pleased with what he saw.
"He has an extraordinary desire to be a protagonist," Montella said.
"He doesn't have the physical sharpness he needs yet, so I'll help him settle in gradually."
