The attacking midfielder is not known for his goalscoring ability, but twice pegged Genoa back after Alberto Gilardino and Luca Antonini netted for the visitors.
Aquilani reached a personal milestone when giving fourth-placed Fiorentina the lead for the first time shortly after the break, but Sebastian De Maio scored midway through the second half to rescue a share of the points from a 3-3 draw.
Vincenzo Montella's Tuscans therefore missed the chance to close to within a point of third-placed Napoli, while Genoa remain without an away league win since November and sit 11th in the table.
Gilardino got the ball rolling against his old team, who enjoyed the first solid chance of a frenetic match.
Joaquin squeezed the ball through to Alessandro Matri, but the former AC Milan striker could not win a battle of wills with Mattia Perin.
Neto then did well to anticipate Gilardino's looping header before Juan Cuadrado set up Massimo Ambrosini to shoot straight at Genoa's goalkeeper.
Fiorentina old boy Gilardino hurt the home faithful in the 27th minute after Luca Antonelli went down cheaply under the attentions of Neto.
The Brazilian keeper was far from happy with referee Dino Tommasi's decision to award the penalty and there was little he could to keep Gilardino's powerfully-taken spot-kick out of the net.
Five minutes later, Fiorentina were given the opportunity to draw level after De Maio felled Ambrosini in the Genoa box. Aquilani did not miss from 11 metres.
However, as soon as Genoa restarted the game, Marvin Compper failed to check Matuzalem's run and the Brazilian midfielder picked out Antonini to restore Genoa's lead.
Eight minutes passed by without another goal being scored before Aquilani amended the situation, completing an early brace with a cool right-footed finish.
The fun continued after the break as Cuadrado embarked on an audacious dribbling run past Moussa Konate before unleashing a drive that De Maio bravely deflected out for a corner.
Aquilani could have scored his hat-trick goal in an incident-packed 53rd minute - the midfielder sliced Manuel Pasqual's pass wide - but he instead did so a few minutes later.
Another lightning-quick Viola raid saw Joaquin skip past Andrea Bertolacci to serve the ex-Liverpool man with the perfect pass and he gladly notched his third goal.
Genoa substitute Ioannis Fetfatzidis soon dazzled the Franchi crowd by wriggling free of Aquilani's marking to steer a left-footed effort just over the bar and, in the 77th minute, the Rossoblu made it 3-3.
When Fetfatzidis lofted his corner across the box to Antonini, the right-back headed the ball across to De Maio, who tucked it away with his second touch.
Neither side was prepared to settle for the draw and Fiorentina perhaps pushed hardest for the win.
Two chances went begging, however.
First, Matias Fernandez slightly misjudged his volley on Anderson's knockdown to send the ball soaring into the stands.
Then, deep into stoppage time, Facundo Roncaglia seized on David Pizarro's pass to fizz it inches past the Genoa goal, capping a dramatic evening with one last hair-raising moment.