However, the Cherries came close to falling behind themselves in a whirlwind opening to the game that saw Jordan Obita and Kaspars Gorkss both go close for the visitors within the opening two minutes.
Once Bournemouth took the lead, they looked more likely to score again when Grabban saw his effort saved by McCarthy.
Moments later, the hosts did extend their advantage when Ritchie took aim at the Hoops goal before finding the bottom left corner of the net.
Bournemouth came close to putting the game beyond the Royals with Kermorgant and Grabban twice going close to scoring a third.
On the stroke of half time, Bournemouth appeared to put the result beyond doubt when Kermorgant headed Simon Francis's cross into the net.