Southampton were without Everton-bound winger Tyler Dibling but made an encouraging start with Adam Armstrong, twice, and Shea Charles having chances inside the first 10 minutes.
They suffered an injury setback when Welington was forced off and eventually taken down the tunnel on a stretcher.
However, it was Saints who came the closest to scoring in an end-to-end first half when Jay Robinson hit the post with both his initial effort from Flynn Downes’ cross, and again from the follow-up.
The home side were almost celebrating again early in the second half when Harwood-Bellis met a Mateus Fernandes corner with a powerful header, only to be denied by a brilliant flying save from Stoke goalkeeper Viktor Johansson.
Two minutes later Stoke took the lead when Thomas broke down the left and Million Manhoef’s deflected shot was saved by Southampton goalkeeper Gavin Bazunu only to fall to Baker to allow him to score his second goal of the season from the rebound.
Stoke were reduced to 10 men just five minutes later when striker Mubama was shown a second yellow card for diving following a one on one with Bazunu.
But, despite that disadvantage, Stoke extended their lead when Johansson released Thomas with a quick throw and the Welshman ran through to fire past Bazunu.
In a repeat of their earlier duel, Harwood-Bellis did beat Johansson with a header from another corner with 11 minutes to go but that proved to be only a consolation for Southampton.