Stoke went into the match without eight injured players and had no answer to the pace of Walcott.
Nigel Quashie set up the goal in the 16th minute with a long distance pass to set Walcott free, and the youngster raced away to beat Steve Simonsen with a low shot.
Southampton had plenty of pressure and numerous opportunities but did not find the target again until almost the last kick when Quashie was again the creator, allowing Belmadi to score with a low shot.
Stoke's cause was not helped by the 39th minute sending off of left-back Marlon Broomes.
He appeared to elbow Walcott and was shown a red card by referee Ray Olivier, before being followed to the dressing room by protesting Stoke manager Johan Boskamp.
Southampton threw away chance after chance, with Neil McCann hitting the top of the bar and shooting wide with only the goal keeper to beat, and Walcott being denied by the diving Simonsen.
Stoke had little idea how to break down the Southampton defence and had only one clear chance, in the 23rd minute, when Hannes Sigurdsson saw a shot kept out by the legs of Antti Niemi.
The Potters found the task of fashioning an equaliser after being reduced to 10 men beyond them and Niemi had nothing else to do.
But Southampton manager Harry Redknapp, mindful of his side's capacity to draw matches they had dominated, was worried that history might be repeated.
Southampton's victory was only their fifth in the league this season and keeps them in touch with the Championship pacesetters.
For Stoke, Broomes now faces a three-match ban, while Boskamp can also expect retribution from the Football Association.