Torino, also indebted to goalkeeper Salvatore Sirigu who made a brilliant one-handed save from Christian Kouame late on, moved into seventh place with 53 points, three behind fourth-placed AC Milan who they host in their next game on April 28.
The Luigi Ferraris stadium was barely half full, with no banners or flags on display, as many Genoa fans boycotted the game in protest over club president Enrico Preziosi's running of the club.
Preziosi said on Friday that he wanted to sell the club and shared the fans' frustration. "I am as fed up as everyone else but there haven't been any serious offers," he said.
Torino played in the old European Cup in 1976/77 but have never taken part under the Champions League format.
(Writing by Brian Homewood; editing by Tony Lawrence)
