Grabban’s spot kick three minutes into the second half secured Cherries' victory at Bloomfield Road on a day when the visitors found it tricky to impose their passing game on their opponents.
Blackpool’s direct approach, a patchy playing surface and the work ethic of both sides resulted in a bitty encounter in which possession was rarely retained for long spells.
But Grabban’s spot kick settled matters and the striker admitted afterwards that the three points had been well-earned.
He told the Daily Echo: “Blackpool have gone through a tough time but they were resilient and we always knew we would have to work hard.
“They were hard conditions and the pitch wasn’t the best. It was a scrappy game and not very typical of us as we didn’t really get our passing going, but we adapted.
“I think we ground out the result in the end and showed a different side to our game.
“It was important for us to get back-to-back wins and start building on our good performances.
“We have to build and create a bit of space between ourselves and the bottom lot in the league. Hopefully, we can do that in the next couple of weeks.”
Grabban was scythed down by Tangerines stopper Matt Gilks in the box less than two minutes after the restart.
The front man took the spot kick himself and although Gilks guessed correctly, the ball zipped past him and into the bottom-left corner.
“I saw him coming so I thought if I could nick it round him, I would either get brought down or put it into an empty net,” said Grabban.
“That was our only clear-cut chance and it came as a penalty and I was down to take it.
“You always know there’s a chance a keeper might get a hand to it but I was just happy when it hit the net.
“I thought there should have been another penalty for the one on Ryan Fraser but at least we got one of them.”
Grabban had a change of strike partner with Yann Kermorgant’s heel injury paving the way for Brett Pitman to make the starting line-up for the first time since January 14.
And Grabban, who now has 14 goals for the season, was impressed with the way the former Bristol City man performed.
“I think he showed good levels of fitness for a player that’s not been playing too regularly,” he said.
“He had a good attitude, worked hard and did well.”