Mark Hughes' side took the lead through a freak 12th-second opener but were pegged back and are now winless in seven league games, while the visitors are unbeaten in eight
In windy conditions for the Premier League match at the Britannia Stadium, a clearance from Begovic bounced over opposite number Artur Boruc to give Stoke a remarkable early lead.
Southampton were the better side for large periods, however, and equalised before the break when Jay Rodriguez converted a teasing cross from James Ward-Prowse.
The visitors looked more likely to grab a winner in the second half and also saw two strong penalty appeals turned down - but both sides had to, ultimately, settle for a point.
Southampton are now unbeaten in eight matches, while Stoke's winless league run stands at seven games.
Begovic, who previously represented Southampton's rivals Portsmouth, is the fifth goalkeeper to score in the Premier League and follows in the footsteps of Peter Schmeichel, Brad Friedel, Paul Robinson and Tim Howard.
|VIEW FROM THE BRITANNIA
|By Alan Bond at the Britannia Stadium
"Full-time here and it's once again Asmir Begovic that will grab the headlines for Stoke. The Bosnian keeper's launch upfield after 12 seconds gave his side an early lead. The scenes of celebration around the ground were incredible. Begovic was muted but Stoke were 1-0 up.
"Back came Southampton, though, who were excellent. Bold and creative in their play, it was James Ward-Prowse who shone today. He was always looking to get on the ball and provided an excellent delivery from set plays and from the right hand side. Indeed, it was the youngster who assisted his sides equaliser crossing for Jay Rodriguez to head down and over Begovic in front of the Saints support."
Southampton, meanwhile named an unchanged starting line-up, with striker Dani Osvaldo missing out once again due to a knock.
Many supporters had yet to find their seat when Begovic got the game off to a scarcely believable start, scoring with Stoke's second touch. His long punt upfield became the fourth-fastest goal in Premier League history.
Buoyed by their stunning opener, the hosts applied further pressure and captain Ryan Shawcross spurned a golden opportunity in the ninth minute after a corner had been flicked into his path eight yards out.
Southampton defender Dejan Lovren was guilty of a similarly bad miss three minutes later, heading wide from an inswinging corner with the goal gaping.
Things settled down thereafter as Southampton regained their composure and started to knock the ball around with increasing confidence.
The visitors' gradual improvement was rewarded four minutes before half-time when Rodriguez popped up unmarked at the far post to meet Ward-Prowse's delivery with a header into the ground that looped over Begovic.
Palacios replaced Whelan at the start of the second half and Stoke came close to retaking the lead in the 50th minute through Marko Arnautovic, only for Boruc to beat away the Austrian's powerful effort from 15 yards.
Southampton were soon enjoying prolonged spells of possession and saw a penalty appeal turned down when Nathaniel Clyne's cross appeared to strike Erik Pieters on the arm.
Ward-Prowse then curled a free-kick narrowly over and Mauricio Pochettino's men continued to look the most likely winners as the game entered its closing stages.
Stoke, who saw Robert Huth booked for a cynical body-check on Southampton substitute Gaston Ramirez, did offer more of a threat late on but could not carve out a clearcut chance.
There was controversy in the 89th minute, however, as another penalty shout from Southampton went unanswered despite Shawcross seemingly blocking a free-kick with his hand.
And Ramirez also saw an effort superbly saved by Begovic with the offside flag raised right at the death.