Palacio opened the scoring two minutes before half-time with a precise glancing header, before the forward doubled his tally deep into stoppage time at the end of a hard-fought encounter.
He now has 12 Serie A goals for the season, five in the last five games, and has become crucial since Diego Milito’s injury, netting all of Inter's goals.
Inter boss Andrea Stramaccioni had declared that his side must win the encounter to maintain their hopes of qualifying for the Champions League next season, and the victory sees his side move above Lazio and into fifth spot.
The Nerazzurri came into the match sitting 10 points behind city neighbours AC Milan, but now they are just a point behind fourth-placed Fiorentina and seven behind their rivals.
They were missing several big names - including the suspended Esteban Cambiasso and the injured Yuto Nagatomo, Cristian Chivu, Dejan Stankovic, Diego Milito and Joel Obi - and came under pressure in the first half as keeper Samir Handanovic made stops from Gianluca Sansone, Mauro Icardi and Nenad Krsticic.
But Inter went ahead just before half-time when Palacio nodded Alvaro Pereira's cross past Sergio Romero.
Samp rallied, Handnaovic making another stop from Sansone and from a Pedro Obiang volley, while he made two good stops from Maxi Lopez.
Late in the match Inter started to threaten again on the break as Fredy Guarin blasted over from a good spot, with Palacio getting the second after a fantastic run and finish.