The Sardinian club were barred from hosting fans at the Stadio Is Arenas due to ongoing safety concerns and local supporters were urged to stay at home and follow the game on television.
Any avid watchers would have to wait until midway through the first half before Marco Sau gave the hosts the lead from the penalty spot following a careless foul from Angelo Ogbonna.
The second half, though, was played at breakneck speed as the lead changed hands three times thanks to quickfire goals between Alessio Cerci, Alen Stefanovic, Conti, Mauricio Pinilla and Rolando Bianchi.
Torino eventually lost both the hapless Ogbonna and his defensive replacement Abou Diop to red cards and Conti duly exploited the space to drive in a scintillating winner from distance, just seconds before the final whistle.
There were a few subtle hints that this would be a classic game as Kamil Glik and Andrea Cossu exchanged promising efforts at the outset.
Cossu then drove a free-kick across Jean-Francois Gillet's goal with Luca Rossettini lurking with intent nearby.
Ogbonna, only recently returned from a lengthy injury lay-off, was a liability throughout and he clumsily fouled Sau in the box in the 37th minute.
Gillet did touch the ball as the penalty came towards him but Sau had ignited a frantic contest with his 11th goal of the season.
Cerci would not wait long to equalise, however, the former Fiorentina winger starting a good team move before leaping onto a Matteo Darmian cross to rifle the ball home.
Torino then turned the game on its head in the 54th minute by taking the lead though Stefanovic as Michael Agazzi flapped wildly at a Cerci cross.
Conti was soon allowed all the time and space he needed to nod home the fourth goal of the day when he connected with a Cossu cross from deep.
Pinilla was denied a fine solo goal when Ogbonna took him down at the end of a surging run, the Italy international's dreadful afternoon drawing to a merciful close before the Cagliari striker converted from the spot.
A quick reshuffle saw Torino coach Giampiero Ventura call Diop off the bench and into the fray but the youngster lasted less than a minute before seeing red for an elbow on Albin Ekdal.
The odds were stacked heavily against Torino but they were given a lifeline when Davide Astori pulled Bianchi's shirt to concede the third penalty of the game.
The former Manchester City man wheeled away in delight against an empty away stand after seemingly rescuing a valuable point for the Granata, but the game was not over yet.
Five minutes into stoppage time, Conti took aim from distance, his swerving effort taking a key deflection to bamboozle Gillet and see Cagliari home at the death.