The new manager has recorded four wins from four games since succeeding Maurico Pochettino, with the young midfielder scored the only goal with a first-half header
Southampton went into their friendly against their south-coast neighbours having won all three of their pre-season matches since Koeman took over from Mauricio Pochettino last month.
And the Dutchman made it four friendly wins from four outings with a 1-0 victory at Dean Court.
Young midfielder Lloyd Isgrove - a graduate of Southampton's renowned youth academy - proved the difference, as his superb first-half header from Nathaniel Clyne's pinpoint cross was enough to beat Eddie Howe's Championship side.
New signing Graziano Pelle started and was lively for Southampton, and he went close to doubling the visitors' advantage shortly after Isgrove's 20th-minute opener.
Koeman's team - shorn of centre-back Dejan Lovren, who is reportedly set to undergo a medical at Liverpool on Saturday - unsurprisingly controlled the vast majority of the game.
However, Bournemouth thought they had equalised late on when Brett Pitman diverted a free-kick beyond Artur Boruc, only for the offside flag to deny them.
And that proved to be the last action of note, with Koeman sure to be eager to make it five wins from five in another south-coast clash at Brighton and Hove Albion next Thursday.