The Cherries blew their visitors away in the first 45 minutes, taking a three-goal lead into half time, with Evanilson opening the scoring with his first goal since arriving from Porto in the summer.
Lewis Cook then saw his goalbound effort deflect in off Dango Ouattara just after the half-hour mark, before in-form Antoine Semenyo notched his third Premier League goal of the season with an excellent solo effort a few minutes later.
The half-time scoreline had Southampton facing the prospect of ending the weekend bottom of the table on goals scored, but they came out for the second period like a team reborn.
Just five minutes after the interval, they had dragged themselves back into the game, when Taylor Harwood-Bellis stooped low to flick a deflected cross beyond Kepa Arrizabalaga.
Bournemouth then struggled to handle threat posed by the away side, who made three changes at the break, including introducing two attackers in the form of Ben Brereton Diaz and Ross Stewart.
Stewart went close with a header shortly after, and Brereton Diaz was unfortunate that nobody got on the end of his square ball that had evaded Kepa across the six-yard area.
Despite the promising signs though, that killer edge was lacking for the most part in the latter stages, as Russell Martin's side fall to another defeat, and remain 19th with just one point from their opening six games.
A first home win of the season for Bournemouth means Iraola's men move onto eight points, and up to 11th, opening up a five-point buffer between themselves and bottom three.
BOURNEMOUTH VS. SOUTHAMPTON HIGHLIGHTS
Evanilson goal vs. Southampton (17th min, Bournemouth 1-0 Southampton)
Excellent quick thinking from Tavernier catches the visiting defence out cold with a free kick chipped in behind, and Evanilson latches onto it to fire beyond Aaron Ramsdale for his first Bournemouth goal.
Ouattara goal vs. Southampton (32nd min, Bournemouth 2-0 Southampton)
Good play from Semenyo down the right creates the opening for Cook who fires goalward from the edge of the area, and his effort ricochets into the back of the net off Ouattara who knew nothing about it.
Semenyo goal vs. Southampton (39th min, Bournemouth 3-0 Southampton)
More stunning footwork sees Semenyo weave his way through a few Southampton players, but despite still being on an angle and having plenty to do, his powerful low drive is out of Ramsdale's reach in the Southampton goal.
Harwood-Bellis goal vs. Bournemouth (51st min, Bournemouth 3-1 Southampton)
Just moments after being denied by Kepa, Harwood-Bellis get on the end of a Matheus Fernandes cross that had inadvertently been flicked on by Ryan Christie, and the central defender stoops low to bring the visitors back into the game.
MAN OF THE MATCH - ANTOINE SEMENYO
It was Semenyo's jinking run that set up Cook's shot for the second goal, and the Ghanaian winger was then able to add to the tally just five minutes later with a stunning solo run and strike, which ultimately put the game beyond Southampton.
BOURNEMOUTH VS. SOUTHAMPTON MATCH STATS
Possession: Bournemouth 41%-59% Southampton
Shots: Bournemouth 14-11 Southampton
Shots on target: Bournemouth 6-3 Southampton
Corners: Bournemouth 6-4 Southampton
Fouls: Bournemouth 19-12 Southampton
BEST STATS
WHAT NEXT?
Another defeat for Southampton is not what manager Martin would have hoped for, especially considering they travel to the Emirates to face Arsenal next weekend, while Bournemouth will look to build on the three points picked up here when they go the King Power Stadium to face Leicester City on Saturday afternoon.