Juventus appeared to be coasting to a fifth consecutive Serie A success when Carlos Tevez and Paul Pogba netted shortly before the interval.
However, the game was turned on its head in a stunning burst after Rossi netted a 66th-minute penalty.
The Italy international then levelled on 76 minutes before a totally unmarked Joaquin fired the hosts ahead moments later.
Rossi then completed his hat-trick, and a barely believable win, when he finished a counter-attack on 81 minutes.
Jorginho scored two second-half penalties as Hellas Verona came from behind to beat Parma 3-2.
Fabrizio Cacciatore put the hosts ahead in the ninth minute before goals from Marco Parolo and Antonio Cassano gave Parma a 2-1 lead after 25 minutes.
Jorginho equalised on the hour mark and the Brazilian then earned and converted his second spot-kick two minutes from time to hand Verona their fifth win of the campaign.
Lazio's struggles in Serie A this season continued as they returned from Bergamo with a 2-1 defeat to Atalanta.
Luca Cigarini put the hosts in front before the interval, but when Brayan Perea levelled during Lazio's brightest spell of the game, Vladimir Petkovic's men looked favourites to go on and find a winner.
However, it was German Denis who scored next, sealing Atalanta's fourth win of the season and taking them above AC Milan and their hosts into seventh.
Substitute Nicola Pozzi's stoppage-time penalty handed Sampdoria their first Serie A win of the campaign as they beat Livorno 2-1.
The hosts looked to have earned a point after substitute Luca Siligardi had cancelled out Citadin Eder's opener with one minute remaining.
However, they then shot themselves in the foot after Andrea Luci brought down substitute Vasco Regini in the area and the referee pointed to the spot.
Pozzi made no mistake and beat Livorno goalkeeper Francesco Bardi to hand his team three vital points.
Alberto Gilardino scored twice as Genoa withstood a second half onslaught to earn a 2-1 win over Chievo.
The three points were vital for both sides as they battled to move away from the relegation zone and it was Gian Piero Gasperini's side who triumphed thanks to Gilardino's double strike.
Gilardino opened the scoring with a header after 22 minutes but Simone Bentivoglio levelled the scores two minutes after the break.
Sassuolo beat fellow strugglers Bologna 2-1 to celebrate their first win of the Serie A season.
Domenico Berardi put the home side ahead in the 12th minute from the penalty spot before team-mate Antonio Floro Flores doubled his team's lead five minutes later.
Bologna's Alessandro Diamanti converted a penalty in the 34th minute to reduce the arrears but the visitors were unable to find the equaliser.
The result has lifted promoted-side Sassuolo to 18th place while it has relegated Bologna to the foot of the standings.
In the late game, Inter Milan moved up to fifth despite conceding a last-minute equaliser in a 3-3 draw at Torino which saw them play for 85 minutes with 10 men.
Samir Handanovic was given a red card after just five minutes with Alexander Farnerud putting Torino ahead (21) only for Fredy Guarin to level on the stroke of half-time.
Ciro Immobile restored the hosts' lead after 53 minutes but that lasted just two minutes before Rodrigo Palacio scored the first of his two goals.
Palacio struck again after 71 minutes and that looked sufficient for victory until Nicola Bellomo's intervention for Torino in the last minute.