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The Potters took the lead inside the first minute as Begovic's wind-assisted kick upfield found its way into the Saints' net.
Stoke should have doubled their lead in the ninth minute when the ball fell kindly for Ryan Shawcross six yards out following a corner, only for the defender to scuff his shot.
Southampton almost snatched an equaliser from a corner in the 13th minute, Dejan Lovren's glancing header sailing just a fraction wide of the post.
The Saints' equaliser came four minutes from the break through Rodriguez, who got his head to a James Ward-Prowse cross to make it all square after 45 minutes.
Southampton came close 10 minutes into the second period from a set piece, Ward-Prowse bending one over the ball and onto the side-netting, leaving Begovic breathing a sigh of relief.
Southampton felt that they should have been awarded a last-minute penalty when Shawcross appeared to block a shot using his arm, but the referee waved away their protests and it remained 1-1.