Edin Dzeko fired Roma in front after quarter of an hour and the hosts dominated for long periods before Icardi struck twice - his third and fourth goals of the season - to stun the home crowd and Matias Vecino completed the scoring.
Inter had welcomed former coach Stefano Pioli back to San Siro with a 3-0 defeat of Fiorentina - his new side - on the opening weekend and it had looked to be Spalletti's turn for an unhappy reunion as Roma cast the visitors aside in a devastating first-half display.
Roma's Radja Nainggolan had earlier been forced to deny claims he had retired from international football after Belgium's bizarre squad snub and the midfielder showed just what Roberto Martinez's men will be missing in their upcoming World Cup qualifiers as he starred and laid on the opener for Dzeko.
Nainggolan was one of three home players to hit the frame of the goal as Roma chased a second, but it was Icardi who netted next against the run of play.
And the Inter captain then scored again with 13 minutes remaining, before Vecino added a third, to secure consecutive league wins in improbable circumstances.
2 - Mauro Icardi has scored 2+ goals in consecutive Serie A appearances for the first time since April 2014. Raging. #RomaInter— OptaPaolo (@OptaPaolo) August 26, 2017
A frantic start saw both sides set off with a great zip to their play, Inter somehow passing up the first clear opportunity as a combination of Icardi and Antonio Candreva failed to bundle the ball into the net from close range.
Aleksandar Kolarov - a scorer on his debut against Atalanta last weekend - then rattled the left-hand post with a low 25-yard volley and, with the breathless pace not letting up, the opener soon followed.
Nainggolan's lofted pass into the area found the chest of Dzeko, whose instant control set up a blistering right-footed volley that left Handanovic with no chance.
Borja Valero miscued a header in a good position as Inter looked to respond quickly, before the same player was fortunate to avoid censure for appearing to throw a senseless elbow towards the brilliant Nainggolan.
Handanovic denied Dzeko a curling second at full stretch and, with time to spare before half-time, Nainggolan's hefty thump struck the same few inches of metalwork that had earlier kept Kolarov out.
Alisson stood firm to parry Icardi's powerful effort at the other end, but the interval did little to slow Roma's momentum and they hit the post for a third time when Diego Perotti so nearly picked out the top-right corner.
However, those small margins proved very costly indeed when Candreva slid a pass through for Icardi to finish beyond Alisson for an unlikely equaliser after 67 minutes.
Dalbert scrambled back to clear off the line when Stephan El Shaarawy's cute dink looked to have immediately restored Roma's lead.
And incredibly, with the game flowing once more from end-to-end, Icardi then found the net again as he spun to shoot low into the net from Ivan Perisic's cross.
Nainggolan had the opportunity to rescue a point late on, but he blazed over from inside the area and Vecino instead put the result beyond doubt from another Perisic centre to leave Roma already three points behind Inter and champions Juventus.
Key Opta stats:
- The previous Inter away win against Roma in Serie A dated back to October 2008 (4-0).
- The Nerazzurri have won each of their last four league matches, scoring 14 times in these games.
- Roma have scored two goals with their very first two shots on target in the current Serie A.
- Edin Dzeko has scored two goals and assisted two in his last four Serie A clashes with Inter.