Samp came into the game, which was initially postponed back in December, full of confidence following Sunday's 3-1 win over Bologna.
But they could not find their range tonight and in the end succumbed to Rafinha's 55th-minute effort.
Indeed, the visitors could well have taken the lead long before they did as they twice hit the woodwork in the first half.
Juraj Kucka struck narrowly over from distance inside the first minute, before, six minutes later, Rodrigo Palacio hit the crossbar from the edge of the box following Kucka's cross from the left.
Samp threatened too, with Eduardo forced to save first with his left foot from Stefano Guberti's close-range shot, then more conventionally when the same player struck right-footed.
At the other end, Kucka shot narrowly wide and Gianluca Curci saved well when Rafinha tried his luck from 25 yards.
Marco Rossi then went close twice, first drilling the ball over after 26 minutes, then going even closer as 10 minutes later he smashed his strike against the crossbar.
Antonio Floro Flores followed up to tap the ball in but he was in a clear offside position.
Samp had the last chance of the half, though, when Daniele Gastaldello headed straight at Eduardo from close range.
Ten minutes after the restart Genoa got the goal they deserved as Rafinha powered the ball into the net from distance.
Rossi then threatened on the counter-attack but no Genoa striker could get on the end of his tempting cross.
At the other end, Guberti fired over from 20 yards, then saw Eduardo deny him from closer in.
But Genoa finished the game in the ascendancy with both Rossi and Kucka going close to adding a second.