The former Rotherham striker scored from the penalty spot early in the second half to secure the Cherries' first away win in five games, but the result piled further misery on the Seasiders.
Saturday's game at Bloomfield Road started slowly with Isaiah Osbourne's high boot, which caught Brett Pitman in the head, the only moment of note in the opening 10 minutes.
Blackpool, who have not won since November, had the first shot of the game in the 11th minute as Stephen Dobbie cut in from the left and fired a fierce effort from 25 yards which was pushed away by Cherries keeper Lee Camp.
Bournemouth had to wait until the 20th minute before unleashing their first shot in anger, but Matt Ritchie's 20-yard drive was comfortably saved by Matt Gilks low to his right.
Ritchie came close again shortly after as Bournemouth grew into the match, running at the Blackpool defence and hitting a left-foot shot that was well held by Gilks.
The bright spark of a first half devoid of attacking flair came on 42 minutes as a floated ball from David Perkins put Dobbie in behind, but his ball across the six-yard box could not find a Blackpool target.
But the visitors broke the deadlock on 48 minutes from the spot - Grabban ran through on goal and was brought down by Gilks just inside the area. Bournemouth's top scorer struck the penalty low to Gilks' right and just beat the keeper to give Eddie Howe's side the crucial first goal.
The away side pushed forward again three minutes later, with Grabban beating Andy Halliday down the right and setting up Ryan Fraser who could not get his shot away.
Blackpool looked to bounce back and came close with successive corners after 54 minutes, with Chris Basham and then Osbourne coming close to an equaliser.
Bournemouth had an opportunity to double their advantage after 68 minutes with a free-kick on the edge of the area, but Pitman fired over.
Blackpool substitute Steven Davies then brought the home crowd to life, winning a contested drop ball before beating three Bournemouth players to win a free-kick 30 yards out, but it came to nothing.
Bournemouth again called Gilks into action on 75 minutes as Ritchie cut in from the right and forced the keeper to dive low to his right to save his left-footed shot.
The Cherries threatened once more through substitute Marc Pugh, his shot curled straight at Gilks, but they held on to leave Barry Ferguson still searching for his first win as Blackpool's caretaker boss.