The Serbian became just the fifth keeper to score in the Premier League when his clearance from his own area was carried the length of the pitch by the wind and bounced over Saints keeper Artur Boruc.
Mauricio Pochettino's side felt that they should have been awarded a late penalty when Ryan Shawcross handled in the box, but nothing was given and the Saints had to settle for a point that extends their unbeaten run to seven top-flight matches.
Below, Sports Mole analyses whether the result was reflective of the action in Staffordshire.
On target: 3
On target: 2
Was the result fair?
Considering that there was an element of fortune to Stoke's goal, and that a couple of decisions could have gone Southampton's way, one would say not. Pochettino's men took a while to settle after the most bizarre start to a Premier League game that you will ever see, but when they did, they handled the conditions fairly well and looked the most likely to grab a winner.
Today's display was evidence of why Stoke are struggling to find the net. Their goal came from their keeper, and only on one occasion did they test Boruc when Marko Arnautovic saw his stinging shot parried away. They matched Southampton's high-intensity, pressing approach and worked admirably from front to back, but they are shot of a creative spark or any sort of service to Peter Crouch from the flanks. Jermaine Pennant and Matthew Etherington are options that manager Mark Hughes may want to explore more thoroughly.
Southampton's performance: It took an extraordinary goal to breach a defence that otherwise looked as solid as they have done throughout a season in which they have only conceded four goals. It was not their best attacking display of the term by any stretch of the imagination, but there was some neat passing moves, with Rickie Lambert and Adam Lallana in particular linking up well. Youngsters Luke Shaw and Ward-Prowse were impressive again.
Sports Mole's man of the match:
James Ward-Prowse: Another promising youngster to have been produced by the Southampton academy, the 19-year-old put in a very mature display and was not afraid to take on responsibility when the visitors were pushing for a winner late on. His delivery from the flanks, or from set pieces, was almost always dangerous, and he came close with a free kick that landed on the roof of the net.
It was mostly luck and the conditions that contributed to Begovic's goal, but Artur Boruc got himself in a bit of a mess when the clearance from his opposite number bounced over him and into his net. It won't go down as a howler, but it will be a moment that haunts the Pole for a long time.
The main talking points regarding Chris Foy's performance today would be his decision not to award the Saints a penalty, amid two handball claims. Foy got the first one right, as Erik Pieters did not seem to deliberately block Ward-Prowse's cross with his arm, but there seemed to be more intent about Shawcross's intervention in the final stages. Well, Southampton certainly thought so.
Stoke: A trip to Swansea City, whose morale may be dependent on the result of tomorrow's first ever all-Welsh Premier League clash against their rivals Cardiff.
Southampton: A fourth-round Capital One Cup tie with Sunderland on Wednesday, before the visit of the Black Cats' conquerors from today, Hull City, at the weekend.