Burnley manager Sean Dyche said they "need players" after Danny Ings' early goal earned Southampton a first top-flight victory at Turf Moor for almost 50 years and ended the visitors' losing start to the season.
Former Clarets forward Ings struck in the fifth minute, supplied by Che Adams's cutback following an excellent through ball by Kyle Walker-Peters.
Injury-hit Burnley, without seven senior players, had claims for a penalty dismissed by referee Andre Marriner - and the video assistant referee - midway through the first half after striker Chris Wood went down under pressure from Jan Bednarek.
Wood went close to an equaliser immediately after half-time but Southampton goalkeeper Alex McCarthy was equal to his low header from Ashley Westwood's cross.