Andrea Stramaccioni's side recovered from a disastrous start to snatch victory at the death, with the Argentine's late strike securing all three points.
The hosts raced into a 2-0 lead thanks to goals from Gonzalo Bergessio and Giovanni Marchese, before Ricky Alvarez reduced the deficit just after halftime.
Palacio, whose introduction inspired Inter's second-half revival, got the equalizing goal before striking for the winner deep into added time.
Coach Andrea Stramaccioni was without a number of first-team players, with Italy international Antonio Cassano dropped for the trip following a dispute between the pair. His absence was felt in a notably sluggish start to proceedings for the Nerazzurri, who struggled in all facts and especially in defense, as Bergessio's half-volley was hurriedly blocked by Juan Jesus. The Rossazzurri would soon be in the lead, though.
An error from Jesus, who attempted to safely see the ball out of play, instead saw him dispossessed by Bergessio, who proceeded to swivel and fire past Samir Handanovic to hand Rolando Maran's side an early lead inside seven minutes.
The visitors struggled to deal with the Sicilians' bright counterattacks, though Zdravko Kuzmanovic's strike from distance only just cleared Mariano Andujar's post. Moments later, however, Inter was 2-0 down to compound a rather embarrassing opening.
A free kick delivered in by Francesco Lodi picked out Marchese in acres of space in the penalty box, with the defender calmly glancing his header past Handanovic and into net to double his side's advantage.
Inter was largely lethargic up front, with Tomasso Rocchi - making his first start for the club - distinctly isolated up front, and it could have been a three-goal edge for Catania as Bergessio spurned an angled strike.
Stramaccioni then brought off the ineffectual Rocchi and Kuzmanovic, and sent on Palacio and Dejan Stankovic.
Palacio's impact was telling almost immediately, as the Nerazzurri emerged a much improved side after the break. Ezequiel Schelotto and Fredy Guarin were both denied, before Palacio's superb run and cross saw fellow Argentine Alvarez loop a header past Andujar to begin a dramatic revival.
The impetus was clearly with the visitors, who looked a different team as the second half wore on, though Catania continued to pose a danger as Bergessio failed to connect with a ball up against Handanovic, an incident which the hosts would rue minutes later.
Alvaro Pereira held off his marker down the left flank and found Palacio with a fine cross, who got in front of the defender to head past Andujar and complete a remarkable turnaround on 70 minutes.
Esteban Cambiasso then volleyed narrowly wide, but there was still to be one last twist in this encounter, deep into injury-time. With 93 minutes on the clock, Palacio turned in from the midfielder's cut-back to steal victory and consign the hosts to just their third home defeat this season.
Inter remains in the race for the Champions League following its late heroics, and travels to London to face Tottenham in the Europa League on Thursday. Catania missed the chance to leapfrog Stramaccioni's side and remains in seventh in Serie A.