Slattery could have put the visitors ahead just before half-time but had a penalty saved by Stoke goalkeeper Jakob Haugaard.
But Hesketh gave Southampton an early second-half lead with a shot from 20 yards and Slattery made amends four minutes later.
Southampton stay ninth in the Division 2 table, nine points outside the play-off places. Stoke are 11th and winless in six matches.
Stoke: 5 Feb v Middlesbrough (H) PL2
Southampton: 5 Feb v West Brom (H) PL2