The 23-year-old striker, on loan from Genoa, struck twice in the space of nine minutes to put his side in control, before Sebastian Giovinco gave Parma a chance in the second half.
The home side did their best to get back into it further, but were unable to get the better of Michael Agazzi in the Cagliari goal.
Giovinco laid on the first chance of the game, crossing for Antonio Candreva who headed wide.
Andrea Cossu then failed to hit the target for Cagliari, before Acquafresca scored his first goal after 22 minutes, drilling in low from the edge of the area, getting the better of Antonio Mirante.
Nine minutes later he was at it again, thumping a left-footed drive beyond Mirante after a ball over the top from Cossu.
Giovinco broke into the Cagliari area before the break but was smothered by Agazzi, with the keeper then blocking Christian Zaccardo's effort.
Mirante stopped Daniele Conti from making it three after the restart, before Giovinco got Parma back into it in the 54th minute, leaving Agazzi stranded with a left-footed hit from the edge of the area.
Hernan Crespo was denied by Agazzi as Parma came on strong, the veteran striker then being denied again by the stopper, before Mirante was called into action again as he beat away Nene`s drive.
Raffaele Palladino thought he had earned Parma a point with four minutes to go but his strike was hacked off the line by Davide Astori, before Mirante denied Alessandro Matri with the last act of the game.