After Oldham's Keigan Parker missed a clear early chance, Hartlepool began the rout after 21 minutes with Brown's sweet finish from the edge of the box. Andy Monkhouse then supplied a deep cross to McSweeney, who headed in from close range with home keeper Dean Brill at fault.
Oldham threatened a fightback early in the second half and Pawel Abbott and Jon Worthington wasted openings but the game was killed off in stunning fashion.
Behan took aim from fully 30 yards with a 61st-minute free-kick and his thunderbolt shot flew into the top corner. He almost repeated the trick from similar distance before sub Colin Larkin was denied by Brill when he looked sure to make it 4-0.
Hartlepool have yet to concede a goal in four league and cup games away from home this season, Scott Flinders protecting his clean sheet when he tipped Alan Sheehan's drive against the post.
Pools moved up to 11th in the table, while Oldham are down to 13th spot.