Kaoru Mitoma drew Brighton level after three minutes of the second half and the visitors seemed the more likely to kick on for a period after their goal.
But the Cherries steadied the ship – and after Antoine Semenyo scored from the penalty spot shortly after the hour mark, they never really looked like squandering their lead.
Match report:
Boss Andoni Iraola was pleased with how his side recovered from that difficult period, and how they then saw the game out.
He told the Daily Echo: “I think we started [the second half] quite well. The first, I don't know, one, two minutes, I think we had a free kick. Brooksy, he didn't probably finish his best shot and from there you concede in the next play.
“I think it's a great goal. I think [Yankuba] Minteh it’s a lovely cross. Mitoma arriving second post in this spot where it's difficult for the keeper, it's difficult for the right back.
“I think we have to value the quality they have. I think especially the wingers are up there like this. It's difficult to find a better pair of wingers in the Premier League and it's normal against Brighton.
“I think we've suffered at the beginning of the second half, but we managed to limit their offensive volume and I'm really happy.”