Aironi had led 10-0 before the Scarlets came back, Stoddart's first score helping them to a 13-10 half-time lead.
The game remained in the balance as Gabriel Pizarro put the hosts back in front, but Nigel Davies' charges held their nerve and pulled clear in the closing stages, with Stoddart rounding off the win.
Tito Tebaldi kicked the hosts in front from an early penalty but missed three more before the scrum half's try and subsequent conversion put his side 10 points up after 27 minutes.
Scarlets fly-half Rhys Priestland had also missed the target with a ninth minute place-kick but was successful four minutes after Tebaldi's try to cut the deficit.
And winger Stoddart touched down four minutes before half-time, Priestland converting and adding a drop goal in the last minute of the first half to send his side into the interval with a three-point advantage.
Another Priestland penalty stretched the margin to six points - but that was wiped out as centre Pizarro went over in the 56th minute and Tebaldi converted to put Aironi 17-16 up.
The Welsh side's response was almost immediate as Priestland added another three points on the hour mark.
That strike settled their nerves and three tries in the final 11 minutes ensured the Llanelli-based side would complete an ultimately comfortable win.
Flanker Johnathan Edwards and centre Gareth Maule both crossed before Stoddart added his second a minute from time, Priestland missing all three conversions to complete a mixed day for the kickers.
The result saw the Scarlets bounce back from last weekend's narrow defeat to Munster and complete a season's double over the Italian newcomers, having beaten them 49-10 at Parc y Scarlets in September.
|Aironi Rugby Score Card|
|Scarlets Score Card|