Mark Hughes was left frustrated as his side failed to finish a plethora of opportunities, with Peter Crouch and Jonathan Walters missing gilt-edged opportunities from close range
Peter Crouch struck the post and Jonathan Walters missing a gilt-edged chance from six yards in the first half for Mark Hughes' men.
Wilson Palacios featured in the Stoke midfield and Asmir Begovic was selected ahead of Jack Butland in goal, while Genoa named Italy international Alberto Gilardino in their starting XI.
Stoke were patient in their build-up play in the early stages and were almost rewarded in the 10th minute when Walters' cross from the right-hand side was met by the head of Crouch but the 32-year-old saw his effort come back off the post.
The home side continued to search for the opener and Charlie Adam shot wide from the edge of the area before Walters should have broken the deadlock.
Crouch knocked a header back across goal into the path of the Republic of Ireland international but he somehow failed to beat Mattia Perin from six yards out.
Hughes introduced several players into the action at half-time, with Glenn Whelan, Cameron Jerome and Jermaine Pennant all entering the fray.
The changes hindered the hosts' momentum slightly as just 30 seconds after the restart Andrea Bertolacci forced Begovic into a fingertip save.
However, Stoke were seldom threatened and Jerome's speculative volley with 20 minutes remaining almost won it for them.
Hughes' side begin their Premier League season with an away trip to face Liverpool on August 17, while Genoa kick off their Serie A campaign against Inter eight days later.