Four minutes Milito again tried his luck on the Sampdoria goal; this time he was denied by goalkeeper Sergio Romero after getting the better of two defenders.
Much against the run of play, Samp broke ahead in the 20th minute when Gianni Munari tucked in a loose ball from close range following a defensive mix-up from a corner.
However, the Nerazzurri continued to dominate proceedings, with Fredy Guarin shooting wide from a good position.
Inter could have restored parity on the stroke of half-time had there been anyone to take advantage of a Palacio cross across the face of the Samp goal.
The hosts did equalise though, and the goal came six minutes into the second half.
Andrea Costa brought down Milito inside his own 18-yard box and was promptly sent off, and the Argentine striker himself strolled home the spot kick.
Palacio was once again making himself available to create chances for Inter and himself went close on two occasions after the hour mark.
The 30-year-old Argentina international deservedly etched his name on the scoresheet on 69 minutes when he shot home from close range after being released by Antonio Cassano.
Guarin added the cherry on the cake for Inter when he made it 3-1 to the Milan giants with a ferocious shot from the edge of the box, although Eder scored for the Genoa outfit in the fourth minute of injury.