Evanilson, Dango Ouattara and Antoine Semenyo struck before halftime for the Cherries to take control of the game. Taylor Harwood-Bellis headed home a consolation for Saints after the break.
Cherries coach Tommy Elphick said afterwards: "It was a game we built up to be quite big beforehand, we've put in quite a few performances this season and not got the points we've deserved so far, so I think the first half was definitely a representation of what we've been doing in spells.
"The tide turned a little bit in the second half, but Southampton are a good team and a tricky team.
"I thought it was a real squad effort."
On Evanilson, he added: "He's a very different number nine to what we had last season. He plays in between centre-backs, he's very cute with his timing of runs, and I'm really pleased for him. He's been working really hard."