Defeat leaves Pools still eight points adrift at the foot of the table, with the Cherries sneaking into the last play-off spot as a result of their comprehensive display.
After Grabban fired over after a short-corner routine and Tommy Elphick headed over after eight minutes, Cherries' keeper Shwan Jalal denied Pools' attacker Jon Franks when he was played in on goal.
And they took the lead just before the half-hour mark when Josh McQuoid headed in firmly from close range when he met an inviting Simon Francis cross.
Ritchie Humphreys wasted a free header to level before Pools did equalise six minutes after the break. Evan Horwood's curling free-kick deceived Jalal, hit the bar and bounced down for Jack Baldwin to turn in his first career goal from close range.
In the 63rd minute the Cherries won it when Sam Collins tripped Eunan O'Kane in the area and Grabban converted to sentance Hartlepool to a seventh successive home defeat.