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.
