However, a second-half resurgence saw the Cherries throw the kitchen sink at Koeman and co, but not even Victor Wanyama's red card 12 minutes from time saw the hosts falter defensively.
Here, Sports Mole analyses how the points were won.
On target: 2
On target: 2
Was the result fair?
Yes. As encouraging as Bournemouth's second-half display was, they did nowhere near enough to test Maarten Stekelenburg and Southampton's defence. It was a game of two halves and, with both sides managing just two shots on target, it is clear that ruthlessness was the difference in the end.
During his pre-match press conference, Koeman had suggested that the visit of Bournemouth could prove a trickier evening for his side than Liverpool and Chelsea had done in recent weeks. Within minutes of kickoff, though, it was clear that Southampton were a level above their South Coast neighbours.
It was a dominant first half an hour, and the opener never looked far away. It came on 31 minutes when Pelle released Ryan Bertrand, whose brilliant cross was volleyed home by Davis at the back post. Pelle then easily outmuscled Sylvain Distin to head a second past Adam Federici. The big Italian joined Southampton's New Zealand-born physio in a makeshift Haka to celebrate. It was a good weekend for the World Cup-winning All Blacks, and barring a miracle, the Saints would be picking up a victory of their own.
Surprisingly, it was their defence which came under the spotlight throughout a second half in which the Saints were dominated. Southampton were given a dose of their own medicine, with Koeman's men more or less camped in their own box for much of the second period. Still, even without Wanyama - who fouled Lee Tomlin on 78 minutes to earn a second yellow card - they were comfortable and never really looked like conceding.
On a weekend where Freddy Krueger claws and Scream masks were the norm, there was no disguising Bournemouth's lack of killer instinct here. Striker Glenn Murray barely touched the ball in an opening 45 minutes in which the visitors were well and truly dominated. It was woeful from Bournemouth and manager Eddie Howe knew it. He replaced Murray and Marc Pugh with Joshua King and Junior Stanislas. It worked.
The Cherries were much better after the break, but for all their possession - which was 74% in their favour at one stage - and momentum, there was no cutting edge, no clear-cut chances and, crucially, no Callum Wilson to pull something out of nothing. Wanyama's dismissal briefly restored some hope, but it never looked like being a watershed moment in a prospective comeback. Harry Arter, on his Premier League debut, came to the fore in the second period but, beyond that, there were few positives for Howe and co.
Sports Mole's man of the match
Graziano Pelle: Another classy performance from the big Italian, whose elegance is seldom seen in a man of his size. His beautiful deft flick-on to Bertrand led to Southampton's opener, before he reminded Bournemouth of his sheer power when he casually brushed Distin out of the way before nodding home the hosts' second.
Due to his Portsmouth roots, the Southampton faithful ensured that Distin endured a torrid time, and the cacophony of abuse appeared to affect the veteran Frenchman. He hardly covered himself in glory for Southampton's first, but it was their second which will surprise Bournemouth boss Howe when his centre-back was far too easily outmuscled by Pelle, who simply nodded home.
Craig Pawson had no choice but to send off Wanyama with 12 minutes remaining, with the already-booked Kenyan bringing down Tomlin. Other than that, Pawson had very little to do other than reject one or two half-hearted penalty appeals in either half.
Southampton: The Saints travel to the North-East next Saturday for a trip to Sunderland.
Bournemouth: Howe and co welcome Newcastle United to Dean Court next Saturday.