The result has moved the Seagulls into fifth spot in the Premier League table, just one point behind second-placed Manchester City, while Bournemouth have dropped to 13th, boasting 15 points from their first 12 games of the season.
The breakthrough came in the fourth minute of the contest through Pedro, with the Brazilian finding the back of the net from close range after Bournemouth goalkeeper Kepa Arrizabalaga had kept out a fierce drive from Georginio Rutter.
The response from Bournemouth was strong, with Brighton goalkeeper Bart Verbruggen required to keep out a strike from Justin Kluivert, before the Dutchman missed the target with another attempt shortly before the 30-minute mark.
Evanilson did have the ball in the back of the Brighton net in the latter stages of the first period, but the offside flag was raised, and a subsequent VAR check showed that it was the right decision, with Brighton entering the half-time break ahead.
The Seagulls doubled their lead in the 49th minute of the match, with Pedro the provider, as his excellent pass was slotted into the bottom corner by Kaoru Mitoma to make it a long way back for the home side.
Brighton were reduced to 10 men on the hour, with Carlos Baleba given his marching orders after picking up a second yellow card for a late challenge on Milos Kerkez, but Bournemouth were unable to score until the 93rd minute through David Brooks, which ultimately proved too late to potentially launch a comeback on home soil.
BOURNEMOUTH VS. BRIGHTON HIGHLIGHTS
4th min: Bournemouth 0-1 Brighton (Joao Pedro)
Brighton take the lead in the fourth minute of the match, as Pedro turns into the back of the net from close range after a powerful strike from Rutter had been pushed by Kepa into a dangerous area.
49th min: Bournemouth 0-2 Brighton (Kaoru Mitoma)
Brighton double their lead in the 49th minute of the contest through Mitoma, with the attacker brilliantly finding the bottom corner after being released by Pedro, who has been a real problem for Bournemouth in this contest.
60th min: Brighton red card (Carlos Baleba)
Brighton are reduced to 10 men on the hour, as Baleba is handed a second yellow card for a foul on Kerkez.
93rd min: Bournemouth 1-2 Brighton (David Brooks)
Bournemouth get one back in the latter stages through Brooks, who finishes off a fine team move.
MAN OF THE MATCH - JOAO PEDRO
Pedro was on hand to send Brighton ahead in the fourth minute with a true centre-forward's goal, and he then provided a wonderful assist for his side's second of the match early in the second period.
The forward was actually replaced in the 65th minute, shortly after Brighton had been reduced to 10 men, but the 23-year-old was excellent at the Vitality Stadium, also boasting a pass success rate of 75%, including two key passes.
BOURNEMOUTH VS. BRIGHTON MATCH STATS
Possession: Bournemouth 55%-45% Brighton
Shots: Bournemouth 19-6 Brighton
Shots on target: Bournemouth 6-4 Brighton
Corners: Bournemouth 7-0 Brighton
Fouls: Bournemouth 12-10 Brighton
WHAT NEXT?
Brighton will be aiming to make it three straight wins in the Premier League when they welcome Southampton to the Amex on November 29.
Bournemouth, meanwhile, will have the chance to bounce back from what was a frustrating defeat when they had to Molineux next Saturday to tackle Wolverhampton Wanderers.