Trailing 3-0 after 16 minutes played, the Italians showed plenty of fighting spirit to launch a comeback in the Europa League Group K fixture.
Emilson Cribari's own goal had put the hosts in front after just two minutes played.
Cristian Tanase doubled Steaua's lead in the 11th minute before Pantelis Kapetanos made it 3-0 shortly after.
Katepanos turned from hero to villain within minutes as he was shown a red card shortly after the half-hour mark.
The Italians made use of their numerical superiority and reduced the arrears with Luigi Vitale scoring on the stroke of half-time.
Napoli substitute Marek Hamsik made it 3-2 with 17 minutes remaining before Cribari stunned the hosts with seconds to go.
Beaten 4-1 in their group opener by Liverpool, Steaua - under new coach Marius Lacatus - produced a much-improved performance at the Stadionul Ghencea.
The home side, looking for their first ever win against Italian opposition, could not believe their luck when Cribari deflected Iulian Apostol's strike into his own net, with Napoli goalkeeper Morgan de Sanctis unable to get a touch.
The Romanians pushed forward and got their second goal of the evening in the 11th minute when Tanase's curled free-kick beat De Sanctis.
Napoli had little time to react as Steaua struck again soon after. Kapetanos latched onto Eder Bonfim's cross and his six-yard strike went into the back of the net.
However, the Italians were given a helping hand when Kapetanos was dismissed after tangling with a Napoli defender.
Vitale handed Napoli a glimmer of hope when he struck from 12 yards to reduced the deficit on the stroke of half-time.
The Italians dominated possession in the second half and were rewarded in the 73rd minute by a fantastic goal from Hamsik.
The Slovakian international midfielder's 30-yard bullet strike beat Steaua goalkeeper Cristian Tatarusanu.
And deep inside stoppage time, Napoli's comeback was complete when Jose Ernesto Sosa's cross found Cavani inside the area - and his right-footed strike from the heart of the area found the back of the net.
Source: DSG
