The Bosnia international, assisted by a considerable tailwind, hoofed the ball downfield when Saints conceded possession from kick-off and was stunned to see it bounce over opposite number Artur Boruc for his first career goal.
But Rodriguez ruined his career landmark moment with a header just before half-time which ballooned off the turf and over Begovic to seal a 1-1 draw.
Nevertheless, Stoke's goalkeeper can still lay claim to the joint-fourth fastest Premier League goal in history, 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 Erik 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 Morgan Schneiderlin and Dejan 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 Artur 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.
The after-effects of the goal appeared to smother the Britannia as both sides struggled to create anything like a clear-cut opportunity.
Jon Walters flashed a cross through the penalty area which evaded everyone while Rodriguez blazed over at the other end.
Southampton have not got to where they are in the table, fifth, without displaying a considerable amount of skill and resilience.
Those traits have produced a remarkable turnaround in the last 12 months as a year ago they were bottom of the table after 10 matches with just four points having conceded 28 goals.
They called on all of those qualities to drag themselves level three minutes before half-time.
James 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.
That was just the lift the visitors needed and, after managing to avoid conceding a goal in the opening moments of the second half, they began to impose themselves on the game more.
The more they pressed the more Stoke's outlet ball was restricted and the home side were gradually forced back.
Ward-Prowse curled a free-kick on to the top of Begovic's net but that was the closest they came to threatening in the 25 minutes immediately after the break.
An angled shot from Marko Arnautovic which was palmed away by Boruc was a rare Potters chance as pressure continued to build with them virtually penned inside their own half.
Lovren's drag-back at a Southampton corner deserved a better finish than him drilling wide but it was indicative of the domination Southampton had.
Substitute Oussama Assaidi skewed a shot wide and Arnautovic's long-range effort was fumbled by Boruc in the last 10 minutes as Stoke launched a late fightback.
Not to be outdone, Begovic ended that match as he started it - in the spotlight - with a brilliant save to deny substitute Gaston Ramirez not knowing the Uruguayan had been flagged offside.
- Sports & Recreation
- Artur Boruc