Francesco Lodi’s deflected strike in the 70th minute proved to be the difference between the two teams as Albion played their third pre-season fixture in the space of six days.
Clear-cut chances were few and far between in the opening stages but Baggies goalkeeper Luke Daniels was called into action midway through the first-half when he tipped Juraj Kucka’s 25-yard effort over the bar.
Both sides made a host of changes after the break, with Steve Clarke handing opportunities to a number of Academy graduates – including midfielder Kemar Roofe.
He came close to scoring just moments after Alberto Gilardino had given Albion a scare when he fired over from a tight angle.
The visitors eventually took the lead after sustained pressure on Daniels’ goal when Lodi’s effort clipped the leg of Shane Long to leave the young goalkeeper helpless to preventing the ball from finding the net.
Clarke handed a debut to another youngster, Wes Atkinson, during the second-half, and the young striker came close to equalising late on but Genoa’s substitute goalkeeper Antonio Donnarumma was equal to his attempt from outside of the area.