The visitors found themselves three-one up at the break, but Matthew Upson's goal midway through the second half set up a tense finale.
A red card for Potters midfielder Steven N'Zonzi did not affect the home side's relentless search for a leveller, with Cameron Jerome's stunner on the 90-minute mark rounding things off.
And, while he admitted the overriding feeling in the visitors' dressing room after the game was one of disappointment, Schneiderlin insisted he and his teammates should be satisfied with their efforts.
"We need to be proud of our performance and our game-plan," the midfielder told Saints Player
at full time.
"I think this is one of the hardest places we will go this year – not many teams take points from this stadium, let alone win.
"We showed that we can change the way we play. We had a game-plan and it worked for 90 minutes, but a superb strike brought them back into the game.
"Before the game, we said a win or a draw would be a good thing but, when you look at the game, everyone is a bit disappointed that we didn't take three points.
"We had two clear chances to put a fourth goal in and I think, if we had done that, it would have been all over.
"We showed both here and at Fulham (where Saints drew on Boxing Day) that we can win games," he summarised.
"We know we can score goals and defend, so we will make sure that we take as many points as we can in the second part of the season."