Icardi was not included in Luciano Spalletti's squad amid an ongoing rift between club and player, with Inter again dropping points in the absence of their star striker after a contentious 3-3 draw at Fiorentina last time out.
Luca Ceppitelli's header gave Cagliari the lead at Sardegna Arena on Friday but Icardi's replacement Lautaro Martinez hit back in the 38th minute.
Pavoletti's excellent effort lifted Cagliari nine points clear of the Serie A relegation zone as the Icardi-Inter row rumbles on, although Alessio Cragno had to brilliantly keep out Martinez with five minutes before Nicolo Barella missed a stoppage-time penalty for the hosts.
Cagliari had lost four of their past five Serie A games but made a bright start with Artur Ionita heading wide before new Inter captain Samir Handanovic brilliantly saved from Pavoletti.
The hosts took a deserved lead in the 31st minute, Ceppitelli beating Ivan Perisic in the air to flick home a dangerous free-kick from Luca Cigarini.
With Inter reeling it should have been 2-0 moments later, Paolo Farago blazing over an open goal after Handanovic made a sensational stop from Joao Pedro's header.
But Martinez had Inter level. Radja Nainggolan's right-wing centre was ideally placed for Icardi's compatriot and deputy to flick a terrific near-post header through Cragno's legs.
Cagliari retook the lead before the break, though, a wonderful cross on the run from veteran right-back Darijo Srna met with a stunning Pavoletti volley that rifled into Handanovic's bottom-left corner.
Cragno saved from Matteo Politano in the 55th minute as Inter sought a fightback, with substitute Borja Valero wasteful and Martinez having a powerful late strike somehow pushed on to the woodwork by the Cagliari goalkeeper.
Rolando Maran's men closed out a memorable victory, though it could have been more comfortable as Barella smashed a spot-kick over the crossbar in the 92nd minute after Milan Skriniar needlessly brought down Kiril Despodov.
Inter dragged into top-four battle
The Nerazzurri seemed certain to finish third until recently but one point from two games is a sorry return that has opened the door to the chasing pack.
Roma and city rivals AC Milan are now within striking distance of Inter as the race for Champions League qualification hots up. Cagliari, meanwhile, are almost assured of safety for another year.
Pavoletti reaches double figures
Prolific Pavoletti was Cagliari's hero for the second home game in a row, the 30-year-old having scored both goals in their previous match at the Sardegna Arena to down Parma 2-1.
Pavoletti has been in the Italy squad on a couple of occasions but a maiden cap is surely now on the cards for the in-form former Napoli man, whose stunning winner was his 10th Serie A goal of the season, ensuring he has now hit double figures in the league two years in a row.