The Brazilian international ignited a rapid-fire three-goal spree for the home team within the first 39 minutes, with Dango Ouattara and Antoine Semenyo also getting on the scoresheet.
Taylor Harwood-Bellis pulled a goal back for Saints early in the second half, but they remain in the bottom three and without a win since their return to the Premier League.
Bournemouth climb above Manchester United into 11th with their second win of the campaign.
Evanilson has big shoes to fill as the replacement for Dominic Solanke, who departed to Tottenham for a club record fee last month.
The £40 million striker did not get off to a good start as he missed a penalty in a 1-0 defeat to Chelsea in Bournemouth's last home game.
However, he banished those memories by latching onto Marcus Tavernier's quickly taken free-kick on 17 minutes.
"It is a goal we've been expecting of Evanilson," said Bournemouth assistant manager Tommy Elphick, who was deputising for suspended boss Andoni Iraola.
"He is so intelligent and makes such cute runs. We haven't been finding him enough so I'm pleased for him."
Ouattara was then in the right place to deflect Lewis Cook's shot past Aaron Ramsdale.
Semenyo has so far filled the void left by Solanke and his fierce drive found the bottom corner for his third goal in six Premier League appearances this season.
It was a familiar tale for Southampton as their dominance of possession counted for little in terms of points.
Russell Martin's men sit second bottom with just one point from their opening six games.
"I am so hurt by the first-half performance after the first goal. I didn't recognise our team," said Martin.
"There was no aggression or courage. They were looking around and in self-preservation mode on the pitch. I didn’t like it one bit. They showed fight in the second half, but it was too late."
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Last season all three promoted sides were relegated from the Premier League for the first time since 1997/98.
The early evidence of this season also suggests the gulf between the Championship and the top tier is becoming increasingly difficult to breach for teams coming up.
Ipswich and Leicester have also failed to register a win in their opening six league matches but do at least sit outside the bottom three.
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