The Royals piled on the pressure in the early stages, twice coming close from corners thanks to the aerial threat posed by Billy Sharp.
Reading controlled the match in the first period, constantly threatening the Cherries defence through Garath McCleary, but they struggled to create much in the way of clear-cut chances.
Bournemouth took the lead against the run of play when Grabban capitalised on sloppy defending from Reading to tap home from close range following a free kick.
The Cherries went into the break 2-0 up courtesy of a strike from Ritchie in the 40th minute as he rifled home from the edge of the area after the hosts had given the ball away cheaply.
Substitute Royston Drenthe inspired some creativity from Reading in the final third, and they pulled a goal of his creation back at the death through Adam Le Fondre, but it proved no more than a consolation.