The centre-back pounced from close range in the 94th minute on Maxi Lopez's flick-on to head the visitors to a second successive top-flight away victory - the first time they have achieved the feat since April 1995.
Lukas Podolski and Nemanja Vidic were the only players to threaten either goal in the first half, the former bring a good save out of Daniele Padelli.
Torino offered more in the final third after the break and brought the first save out of Samir Handanovic after 72 minutes, Cristian Molinaro having his side's first shot on target.
The win lifts Torino to within a point of Inter, who saw their hopes of a top-four finish dented.
Palacio created the game's opening chance after 13 minutes for Podolski, but the Germany international's low shot was parried by the Torino goalkeeper.
Torino's best opening arrived, Josef Martinez inexplicably leaving a Marco Benassi cross from the right when unmarked in the penalty area.
Handanovic was finally called into action with 18 minutes remaining, the Slovenia international getting down well to hold Molinaro's speculative effort.
Icardi wasted the chance to become Inter's hero in the final 15 minutes as he missed Palacio's low cross and Torino eventually cleared to safety.
It was the visitors who finally broke the deadlock when Moretti netted at the death.
Lopez's front-post flick found the 33-year-old unmarked six yards from goal and he made no mistake, much to the frustration of the home fans.
