Eddie Howe's side came into this match having lost six of their past seven Premier League matches yet Brooks' goals in either half helped them stop the rot ahead of tricky ties against Spurs and Manchester United.
Brighton, who had Lewis Dunk sent off, we left cursing their poor finishing after squandering a hat-full of chances.
The Seagulls set the early pressure as Yves Bissouma fired over from the edge of the box after only six minutes.
The Mali international then saw his effort saved by Asmir Begovic after a clever one-two with Pascal Gross.
The Bournemouth keeper was again called into action when he palmed away Jurgen Locadia's low show from the edge of the box.
Yet the Seagulls were silenced when Brooks put the Cherries ahead on 21 minutes when he curled home his fourth goal in seven games.
The Wales international benefited from some poor defending from Brighton as Davy Propper's lung was lacklustre at best, while Leon Balogun turned his back on the midfielder.
And just before the break, Begovic made it a first-half hat-trick of saves when he turned away Dunk's header from Gross's cross from the left.
Junior Stanislas came close to doubling Bournemouth's lead in the 63rd minute with a dipping free-kick that just squirmed wide of Mat Ryan's left-hand post.
It was the winger's final contribution to the match, as Ryan Fraser came on to add to Bournemouth's attacking threat.
And the home side were given a further boost when Dunk was dismissed on 73 minutes for a second yellow card.
The Brighton defender had earlier been booked by referee Mike Dean for fouling Brooks and there were no complaints about his second as he hauled down Callum Wilson.
Bournemouth made their extra man advantage count as Brooks added his second on 77 minutes with a neat backwards header from Fraser's floated ball into the box.
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