The visitors came close to breaking the deadlock after just eight minutes when Angelo Palombo flashed a free kick just past the left post from 19 yards out.
Genoa didn't take heed of that warning shot and found themselves behind midway through the first half as Eder picked Maxi Lopez, who in turn fired an effort into the bottom corner from inside the box.
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The hosts' first real chance of the match came with just four minutes remaining in the first half as Andrea Bertolacci went for goal from the edge of the box, but Angelo da Costa pulled off a fine stop to deny the Italian.
Sampdoria should have doubled their lead immediately after the restart when Roberto Soriano was sent through on goal with only the keeper to beat, but his shot was poor and easy for Mattia Perin to gather.
Perin was called into action again 10 minutes later when he was forced to deny Lorenzo de Silvestri's low effort from the edge of the box.
Lopez almost got a second for the visitors with 15 minutes left when he pulled down a pass inside the box before drawing another impressive save from Perin.
Alessio Sestu was the next to come close for Sampdoria as he drilled a low strike towards goal having been found from a free kick, but the shot was straight at the keeper.