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 arrears just after half-time.
Rodrigo Palacio, whose introduction inspired Inter's second-half revival, got the equalising goal, though the hosts will rue the number of chances they missed, as Palacio struck the winner deep into added time.
Coach Andrea Stramaccioni was without a number of first-team players, not least Antonio Cassano following a dispute between the pair, with the Italy international subsequently dropped for the trip.
And Cassano's absence was felt in a notably sluggish start to proceedings for the Nerazzurri, whose defence struggled from the off, as Bergessio's half-volley was hurriedly blocked by Juan Jesus, though the Rossazzurri would soon be in the lead.
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 were 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 themselves were largely lethargic up front, with Tomasso Rocchi - making his first start for the club - distinctly isolated up front, and it could have been three for Catania as Bergessio spurned an angled strike.
Stramaccioni then brought off the ineffectual Rocchi and Kuzmanovic, and sent on Rodrigo Palacio and Dejan Stankovic.
But the Sicilians continued to create chances to extend their lead, as a fluid break saw Handanovic thwart Bergessio on two occasions before Inter's comeback.
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 remain in the race for the Champions League following their late show, and travel to London to face Tottenham in the Europa League on Thursday. Catania miss the chance to leapfrog Stramaccioni's side and remain in seventh.