Jerome flashed home a sensational 25-yard effort to cap a Stoke comeback from 3-1 down at half-time, a rally all the more impressive considering that they had Steven N'Zonzi sent off with 20 minutes to play.
After a quiet start, the game burst into life in the 10th minute when Southampton made the most of the first real chance, the dangerous Rickie Lambert popping up at the far post to convert a Guly do Prado cross that eluded a number of defenders.
Kenwyne Jones had a good chance to level moments later but he poked a shot over the bar after N'Zonzi's effort had been deflected into his path.
The equaliser did arrive on 16 thanks to a lovely bit of skill from the big striker, who cleverly flicked Ryan Shotton's low cross past Kelvin Davis with his back-heel.
Southampton restored their lead eight minutes later when Lambert's precise cross was knocked on to the bar by a desperate attempted clearance by Robert Huth and fell nicely for Jay Rodriguez to tap home from a yard out.
Their forward pressure paid off again nine minutes before half-time when Lambert's knock-down was diverted past Begovic by Andy Wilkinson, leaving Stoke's previously proud defensive record - just four goals conceded in nine home games - in tatters, although they were without regulars Ryan Shawcross and Geoff Cameron.
Stoke proved more positive early in the second half but, bar a Jonathan Walters shot that was blocked by Maya Yoshida, they struggled to put Davis under much pressure before a lengthy delay caused by a nasty clash of heads between Wilkinson and Huth that left both bloodied.
More defensive hesitation almost gave Southampton a fourth before Huth scrambled back to block a Rodriguez effort with Begovic stranded way off his line.
Begovic redeemed himself by denying Rodriguez a second goal, blocking his effort with an outstretched foot, and do Prado will be kicking himself for missing an open goal when the rebound fell perfectly for him on the volley.
Stoke got themselves back into the game 23 minutes from time when a corner eventually fell to Matthew Upson to squeeze a shot past Davis after Jones' original effort had been blocked.
Seconds later home fans were aggrieved not to get a penalty when Jose Fonte, who had already been booked, appeared to get away with a clear handball in the box after Jones went in for a header.
Their hopes were not helped on 71 when N'Zonzi was shown a straight red card for an over-the-top tackle on Jack Cork, who was lucky the midfielder didn't make too much contact with his knee.
Stoke refused to lie down and forced a series of corners, along with another appeal for a penalty for a possible handball by Yoshida, and they had a Cameron Jerome effort correctly ruled out for offside.
But the equaliser arrived in stunning fashion with 10 seconds left of the 90 minutes when another sub Peter Crouch combined with Jerome, who lashed an unstoppable half-volley past a helpless Davis into the top corner and made it 10 top-flight games without defeat at home this season for Tony Pulis' side.
Crouch came close to snatching all three points in injury time, forcing Davis into an unconvincing save, and Southampton were eventually thankful to scramble the ball clear and leave with a point.