With the Sicilians pushing hard for a Europa League place, Andrea Stramaccioni's men appeared all too happy to oblige their hosts as Catania took a quick two-goal lead through Gonzalo Bergessio and Giovanni Marchese.
That prompted a desperate reshuffle and a gritty Inter side steadily forced their way back into the game as Palacio followed up Ricky Alvarez's 52nd-minute header.
Inter were not prepared to concede fourth place to Fiorentina, who beat Chievo 2-1, and Palacio struck at the end to keep the former European champions within a point of the Champions League positions.
Inter could not have made a worse start, though, as Juan Jesus handed Catania the lead on a plate in the seventh minute.
The hapless Brazilian defender was found wanting with Bergessio lurking, and goalkeeper Samir Handanovic was left totally unprotected as the Argentinian picked his spot.
It got much worse for a poorly organised Inter defence as an unmarked Marchese rose highest to head home a Francesco Lodi free-kick.
The Nerazzurri were brighter when they managed to play the ball out of their own half and Tommaso Rocchi just needed a little more accuracy when sent through by Alvaro Pereira.
Catania had little trouble keeping their opposition at bay and Ricky Alvarez was restricted to a shot from distance when trying to carve his way into the box.
There was a flicker of revival one minute before the break but Fredy Guarin's dangerous low cross was closed down by Pablo Alvarez, with the home crowd cheering their team's every touch.
Inter boss Stramaccioni, who was rightly concerned about his team's shape, made two changes at the break as Palacio and Dejan Stankovic replaced Rocchi and Zdravko Kuzmanovic.
There was time for Handanovic to deny Bergessio in a one-on-one before Inter forced their way back into the game in the 52nd minute. A fine piece of individual skill saw Palacio skip past Alexis Rolin to tee Ricky Alvarez up for a header that beat a stranded Mariano Andujar.
Stramaccioni cut a concerned figure on the touchline as Guarin came off the worse from a collision with Marchese, but the Colombian was soon back on his feet and leading Inter's revival.
His header from a Pereira cross was too high but both men played a key role as the Nerazzurri levelled the tie in the 70th minute. A fine team move saw Pereira deliver a superb cross from the left wing and Andujar was left at sixes and sevens as Palacio rose to head the ball home.
All of a sudden Inter's tails were up and it fell to Alejandro Gomez to race back and deny Walter Gargano at the end of a jinking run. A much better opportunity followed and Ezequiel Schelotto was only denied his second Inter goal by a committed double save from the under-siege Andujar.
There was time, of course, for one last dramatic twist at the Stadio Angelo Massimino and it was written by Palacio two minutes into stoppage time.
Once again the Argentinian was at the heart of the action, collecting a cute pass from Esteban Cambiasso and beating another countryman, Andujar, to cap a remarkable turnaround.