Three goals in 15 minutes made for a frantic first half at the Blue Water Arena, but Aquilani's penalty secured a two-goal win for the visitors.
On-loan AC Milan man Alessandro Matri showed solid skill to score the opener but Danish forward Martin Pusic had equalized a minute later after the Viola failed to clear a dangerous cross.
Five minutes later, though, Josip Ilicic had cooled the visiting fans' nerves as the hosts similarly did not deal with a whipped ball in well enough, and Alberto Aquilani effectively killed off the tie with a converted penalty with halftime lingering.
Vicenzo Montella's men saw out the second half with a little less frenetic goalscoring at either end to leave them with a considerable advantage heading into the latter half of the Europa League tie at the Stadio Artemio Franchi.
German striker and summer signing Mario Gomez was on the bench once again for Fiorentina, having made his comeback from a long spell out injured as a substitute over the weekend.
Pusic will have made Montella question how soon he might need a forward like Gomez on the pitch as Pusic tested Viola shot-stopper Neto with four minutes gone.
But it was the visitors who got the scoring under way. Matri, on a half-season loan at the Florence side, controlled a pass perfectly then buried his effort in the bottom corner.
Before Fiorentina even had a chance to settle following the celebrations, Pusic pounced on a deep cross to fire Esbjerg level with a low and accurate attempt which brought the score to 1-1.
The breathless first 15 minutes ended with Ilicic also scoring from a threatening cross curled into the box and returning the Italian side to the lead.
The tempo died down in the subsequent five minutes and the next chance came with a third of the game gone as Esbjerg midfielder Markus Levken's cross narrowly avoided causing havoc and instead went out for a corner.
Just as the hosts were starting to rediscover their verve — with Peter Ankersen's deflected effort just going wide of the post — Davidson Drobo Ampem brought down Ryder Matos just inside the area to concede a penalty for Fiorentina.
Esbjerg's players protested the foul was outside the box to no avail and ex-Liverpool midfielder Aquilani hit home to give the visitors a comfortable cushion.
Halftime came and went without further major incident, with Mario Gomez's arrival on the pitch for goalscorer Matri spicing up Montella's tactics.
The former Bayern Munich forward very nearly made an instant impact upon his introduction, forcing a save from Martin Dubravka with a shot from the edge of the 18-yard box.
As the second half wound down to its predictable conclusion, Borja Valero should really have given Fiorentina an additional away goal in the European tussle but he screwed a rebounded Ilicic shot over the crossbar, wasting all of Matos' excellent work in creating the chance.
It did not matter with regard to the result, however, and Montella's men can consider themselves as having one foot in the next round after an impressive display at the Blue Water Arena.