The Serie A victory over Vincenzo Montella's confirmed Coppa Italia finalists saw Inter close to within five points of their fourth-placed hosts in the last European berth.
La Viola were off colour following their midweek cup heroics and finally succumbed to Inter's relentless first-half assault when Palacio used close control to outwit goalkeeper Neto.
Juan Cuadrado exploited a lapse in concentration from Samir Handanovic to level immediately after the break but Icardi emerged from the bench to score the winner.
There was definitely a buzz in the air at the Franchi as Viola fans returned to their seats only four days after watching their beloved team overturn Udinese's first-leg advantage to reach the Coppa Italia final.
That excitement did not translate to the players, though, with Montella's men subdued in the opening exchanges, and Inter quickly took the initiative.
Neto looked nervous as he failed to deal with David Pizarro's backpass, the Fiorentina keeper only just getting the ball clear as Palacio came sniffing.
The Argentinian striker came close to opening the scoring when a swift Inter counter saw him hold off the challenge of Marvin Compper to rattle a post with an angled drive.
Diego Milito soon forced a reaction save from Neto when set up by Hernanes and Yuto Nagatomo's header from a Palacio cross was deflected out for a corner.
Inter continued to turn the screw and Milito gave Neto another warning when skipping past Mati Fernandez to smash just wide of the mark.
The opener would come in the 34th minute, Palacio controlling Fredy Guarin's lofted pass on his chest before swivelling and clipping the ball beyond the flailing Neto.
Fiorentina were scrambling and were lucky to avoid conceding again in the moments before the break as Nagatomo shot wide from close range under the attentions of Cuadrado.
Montella gave his players a rocket at the interval and it worked as Cuadrado fired Fiorentina level in the seconds that followed the restart.
The Colombian perhaps benefited from a ponderous response from Handanovic, the Inter keeper leaping over a curling effort that nestled in the bottom corner.
Fiorentina might even have turned the game on its head had Joaquin found the empty net when Handanovic misjudged the flight of Cuadrado's cross.
Inter boss Walter Mazzarri introduced Icardi for Milito in a bid to wrest back control of the match and the move paid dividends almost immediately.
The former Sampdoria forward - without a goal since September - latched onto Nagatomo's cross from deep on the left and volleyed the ball past the badly-positioned Neto to restore Inter's lead.
Montella's response was to reintroduce Mario Gomez after a six-month absence, the German striker making his long-awaited return from injury as he replaced Joaquin midway through the second half.
His fellow substitute Alessandro Matri was heavily involved as Fiorentina came close to grabbing a late equaliser, his flick-on to Josip Ilicic from a Fernandez corner only missing the mark thanks to a brave block from Hernanes.