Bournemouth's Sam Surridge was denied a stoppage-time equaliser before Che Adams added to Danny Ings' 21st goal of the season for Saints.
Danny Ings' first-half strike set Southampton on the way to a 2-0 win over Bournemouth that placed his former club on the brink of relegation.
Ings arrowed in a fine finish to crown a sweeping Saints move in the 41st minute to claim his 21st Premier League goal of the season.
He could have closed to within one of Jamie Vardy in the Golden Boot race but for a tame 59th-minute penalty, saved by Aaron Ramsdale down to his left after Harry Wilson handled following a corner.
Bournemouth thought they had made the most of that let-off with a stoppage-time equaliser from substitute Sam Surridge, only for VAR to show Callum Wilson to be offside and interfering with play, before Southampton replacement Che Adams dished out a little more damage that the Cherries can ill afford.
Eddie Howe's men are three points from safety with only a final-day trip to Everton to come. Survival on goal difference is now the best they can hope for and the end of a five-season stay in the Premier League will be confirmed if Watford – reportedly on the brink of sacking manager Nigel Pearson – are able to take a point from either of their remaining matches versus Manchester City and Arsenal.