Bournemouth have now gone eight matches unbeaten since manager Eddie Howe returned to the south coast as manager from Burnley, while Oldham's defeat was their third in succession.
After a goalless first half Bournemouth stormed in front with Grabban scoring his first two goals in a minute.
The former Rotherham striker got his first in 53rd minute, turning Harry Arter's pass home and seconds later converted a cross from the right by Simon Francis.
Grabban's third came in the 72nd minute after Arter had been fouled in the penalty area. Arter's spot kick was parried by goalkeeper Dean Bouzanis, but Grabban was first to the rebound.
Charlie Daniels headed his fourth goal of the season in the 81st minute before Matt Smith responded for the visitors with a header to earn Oldham a consolation, but they had posed little threat after the break.