Former Genoa striker Palacio scored in the 43rd minute of the match - which had been postponed two weeks ago due to heavy snowfall at the Marassi - and again in second-half stoppage time to move Inter up to fifth - seven points shy of third-placed AC Milan with eight games left.
Samp, meanwhile, slipped one position to 13th but stayed 10 points clear of the relegation zone having enjoyed several promising chances until Inter shut up shop for much of the second half.
The first 45 minutes were dominated by Sampdoria but punctuated with several dangerous Inter raids which would ultimately yield a goal just before the break.
Gianluca Sansone's flying header from a Lorenzo De Silvestri cross showed Samp meant business in the early exchanges before Palacio burst through to win a corner with an angled drive.
The Argentinian then latched onto Antonio Cassano's weighted ball forward to ripple the side-netting, but he had been a yard offside.
Tonight was an excellent chance for Inter's scouts to watch Samp's nine-goal striker - and rumoured transfer target - Mauro Icardi up close and he did not disappoint, first heading just over the bar before Samir Handanovic struggled to save his header from a Sansone corner.
Nenad Krsticic went even closer as he cut inside to test Handanovic with a left-footed effort but Samp would live to regret their missed chances.
In the 43rd minute, Palacio broke forward again, on this occasion timing his run perfectly to nick in between the centre-backs and head Alvaro Pereira's cross into the back of the net.
A quiet restart built steadily until Palacio came knocking in the 56th minute, but Shkrodan Mustafi took the ball off his feet at the critical moment.
The introduction of Maxi Lopez for the struggling Sansone freed Samp up in the final third for a hectic period and it was Pedro Obiang who finally found space to test Handanovic.
Inter's priority, though, was preserving their lead and, when midfielder Zdravko Kuzmanovic replaced Cassano, it was clear the final 20 minutes would not be pretty.
The Nerazzurri rode out several Samp raids until breaking forward at speed two minutes into time added on, with Palacio coolly ending the match as a contest from a one-on-one with Sergio Romero.
