Begovic got the opener after just 13 seconds, with a long kick forward taking one bounce and flying in over Artur Boruc's head, helped by the wind.
Ryan Shawcross then missed a volley with the goal gaping after Jon Walters' flick-on from a corner, while Dejan Lovren headed wide from a corner at the other end.
Saints got more into the game and notched an equaliser shortly before half-time with Rodriguez heading over Begovic from James Ward-Prowse's cross from the right.
In the second period Saints had two penalty shouts against Erik Pieters and Shawcross refused as the game ended in a draw.
Begovic can lay claim to the joint-fourth fastest Premier League goal in history with his first goal, beaten only by Ledley King - himself not prolific by any standards - and more seasoned goalscorers Alan Shearer and Mark Viduka, and on a par with Dwight Yorke and Chris Sutton.
What was more criminal from Southampton's point of view is that they had taken the kick-off and given the ball away.
Full-back Pieters tidied up to his goalkeeper, on his own penalty spot with a strong wind blowing from behind his goal, and he launched it downfield.
Southampton's two centre-backs Jose Fonte and Lovren both stood aside to allow the ball to bounce five yards outside the box but it immediately seemed to pick up pace and looped over Boruc, on his own penalty spot, and into the net - a distance of roughly 98 yards.
There must be something about the Britannia Stadium which lends itself to long-range goals as Wigan defender Maynor Figueroa scored from two yards inside his own half in December 2009 - although that was at the opposite end.
And Southampton are no strangers to being beaten by a goalkeeper as Mark Crossley scored a late equaliser while on loan at Sheffield Wednesday in a 3-3 draw at Hillsborough in December 2006.
Begovic looked almost embarrassed and, if Stoke's players were not sure how to react, Southampton's looked shell-shocked by the freakish goal and it took them a while to regain their composure.
Shawcross could have doubled the lead but missed a volley from close range from a corner, while Lovren headed wide from close range at the far post from a corner at the other end.
But Saints dragged themselves level three minutes before half-time when Ward-Prowse swung a cross to the far post and Rodriguez, while not making the best of contact, headed down and the ball ballooned over Begovic and into the far corner of the net.
Soon after the break, Marko Arnautovic saw his shot on goal from the right well saved by Boruc.
At the other end, Nathaniel Clyne crossed from the right, with the ball striking Pieters on his arm in the box but a spot-kick was not awarded.
Ward-Prowse then curled a free-kick on to the top of Begovic's net before an angled shot from Arnautovic was palmed away by Boruc in a rare Potters chance.
Lovren's drag-back at a Southampton corner deserved a better finish than him drilling wide but it was indicative of the domination the visitors had.
Stoke substitute Oussama Assaidi skewed a shot wide and Arnautovic's long-range effort was fumbled by Boruc in the last 10 minutes as the Potters launched a late fightback.
Begovic made a brilliant save to deny substitute Gaston Ramirez not knowing the Uruguayan had been flagged offside.
And Saints saw a late penalty shout against Shawcross turned down as the sides shared the spoils at the Britannia.