At the San Siro, Palacio struck four minutes from time with a neat backheeled finish from Fredy Guarin's pass to send three quarters of the stadium into a frenzy and Inter back up to fifth, 15 points behind leaders Juve.
The result was tough on Milan, who dominated for long periods and should have led on 30 minutes when Inter keeper Samir Handanovic flapped at a corner and the ball fell to Andrea Poli, who could only blast over from 10 yards.
Inter finally found some rhythm as half-time approached and were denied a penalty when Palacio tumbled under pressure from Cristian Zapata, before Esteban Cambiasso headed wide.
Mario Balotelli twice went close after the break, with his shot on 53 minutes well saved by Handanovic before the former Manchester City striker poked just wide with the outside of his boot.
Again, Inter grew stronger as the half progressed and Abbiati did brilliantly to deny Palaico's turn and shot from six yards.
But he could do nothing to prevent the Argentina forward from stealing the points, his cheeky finish in a crowded area preceding Sulley Muntari's late red card for a needless challenge on Zdravko Kuzmanovic.
Earlier, Juventus beat Atalanta 4-1 to secure their ninth straight win and maintain a five-point lead over Roma, who eased to a 4-0 victory over bottom club Catania.
Juve were set on their way by Carlos Tevez, only for Atalanta to level when Giacomo Bonaventura pulled the ball back for Maxi Moralez to fire in off the inside of the left upright.
Juve were ahead again just after the break - Fernando Llorente touching on Kwadwo Asamoah's cross for Paul Pogba to slide the ball in from close range - before Llorente added a third on 75.
Arturo Vidal added a fourth four minutes later, tapping in Stephan Lichtsteiner's cross.
In the capital, Roma took the lead on 18 minutes when Mattia Destro nodded on a corner for Mehdi Benatia to head in from six yards.
Catania gifted Roma a second goal 11 minutes into the second-half when keeper Alberto Frison's poor positioning allowed Maicon's cross to bounce over him and straight to Destro for a simple tap-in into an empty net.
Benatia extended Roma's lead moments later, heading Adem Ljajic's free kick into the bottom right corner, and Gervinho, having missed a sitter on 75, atoned with the fourth to finish from Ljajic's cross.
Elsewhere, Giuseppe Rossi's late strike was enough for Fiorentina to claim a hard-fought 1-0 win at lowly Sassuolo, while Bologna saw off Genoa by the same scoreline courtesy of Alessandro Diamanti's 57th-minute goal.
Luca Toni scored twice as sixth-placed Verona thrashed Lazio 4-1, while Ciro Immobile was also on target with two goals in Torino's 4-1 victory over Chievo.
Sampdoria extended their unbeaten run under Sinisa Mihajlovic to five games but had to settle for only a point in a 1-1 draw with Parma.