Leading 2-0 from the first leg in Austria, Liverpool lacked cohesion at Anfield but held on for a 2-1 aggregate victory.
Although they didn't look like scoring during the first half, a top class goal from Mario Tokic gave the visitors the lead just after half time.
From then on Graz had good spells of possession and threatened an upset, but Benitez's side are through to the first round of the competition.
