The Welsh side's response was almost immediate as Priestland added another three points on the hour mark.
It remained in the balance as Gabriel Pizarro put the hosts back in front but Nigel Davies' side held their nerve and pulled clear in the closing stages, with Stoddart rounding off the win.
before the Scarlets came back, Stoddart's first score helping them to a 13 10 half time lead.
StretchedAnother 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.
Flanker Johnathan Edwards and centre Gareth Belt Hermes For Man Maule both crossed before Stoddart added his second a minute from time, Priestland missing all three conversions to complete a mixed day for kickers.