It was a third win of the season for the beleaguered Saints, who climbed four places to 14th in the standings.
They had to put up a stiff rearguard action to hold on to their lead, however, as Bournemouth peppered them in the second half and came close to equalising on several occasions.
They also had a penalty appeal against centre back Mohammed Salisu turned down after a VAR check 15 minutes from time while desperate defending kept out Philip Billing and substitute Junior Stanislas as Bournemouth tried to force an equaliser.
It meant Southampton's first clean sheet in 17 games since a 1-0 home victory over Arsenal in April, although goalkeeper Gavin Bazunu dropped a high ball close to his own line in stoppage time in a heart-stopping moment for the away fans.
Southampton did have a chance to put the result to bed eight minutes from the end but Adams had his close-range shot blocked by home skipper Adam Smith.
The result ended an unbeaten run of six games for promoted Bournemouth since the sacking of Scott Parker following a 9-0 drubbing at Liverpool in late August.
Caretaker manager Gary O'Neil had not lost a game in charge before Wednesday and Bournemouth remained in 11th spot.
Southampton host league leaders Arsenal on Sunday while the Cherries travel to West Ham United on Monday.