Roberto Mancini's men had suffered back-to-back defeats in their last two games – part of a run that saw Inter slip to 13th.
The hosts looked the better side at San Siro on this occasion and went ahead through a Fredy Guarin header in the 16th minute, although they did have to ride their luck at times.
Palermo will wonder how they were unable to pull level in the 50th minute, when a remarkable series of goalscoring opportunities culminated in Paulo Dybala blasting over from two yards out.
Inter struck the woodwork twice at the other end before Mauro Icardi doubled the lead with a low drive in the 65th minute, and the Argentine put the result beyond doubt with a headed effort two minutes from time.
The result sees Inter climb above local rivals Milan on goal difference into ninth.
Guarin came into the Inter starting XI as one of three changes to the side that suffered a 1-0 defeat to Napoli in the Coppa Italia quarter-finals on Wednesday and, after Juan Jesus had tested Stefano Sorrentino with an early header, the Colombia international broke the deadlock.
The former Porto man rose highest from a Xherdan Shaqiri corner to head the ball against the ground and beyond Sorrentino.
Dybala had curled over a free-kick from a dangerous position prior to Guarin's opener, and Palermo's first chance to equalise went begging as Luca Rigoni found the side-netting from an Achraf Lazaar cross.
Yuto Nagatomo was forced from the field after receiving treatment for an apparent thigh injury, with Dodo coming on in his place, and Inter struggled to test Sorrentino after going ahead – Gary Medel's optimistic long-range strike about as close as they came before half-time.
Palermo should have been level five minutes into the second half, but a series of glorious opportunities failed to bear fruit.
Edgar Barreto squared to Dybala when shooting was arguably the better option, before the ball broke to Franco Vazquez on the edge of the penalty area.
The 25-year-old's low drive hit the left-hand upright but bounced across the face of goal to Dybala, who inexplicably blasted over from point-blank range.
Inter hit the woodwork twice at the other end soon after – Icardi sending a clever looping header against the right-hand upright, before Andrea Ranocchia headed against the crossbar.
Mancini's side doubled their advantage in the 65th minute, though, when the ball found Icardi in space inside the Palermo box, and the Argentine converted a low strike.
Both goalscorers combined for Inter's late third - Guarin chipping in a cross from the byline for Icardi to head home and make sure of three points and a position above Milan in the table.
