A South African penalty in front of the posts was turned down and it proved an inspired decision as flanker Burger charged over for a try in the 16th minute after some quick hands from Morne Steyn and Wynand Olivier.
The controversial Burger's reaction to his score did not endear him to the home crowd as he thumped the ball into the stands.
Indeed, the suggestions of needle between the two squads arising from the summer Lions tour seemed to be coming to pass early on as afters was the order of the day on a number of occasions.
The Irish responded by putting the South Africans under intense pressure as Jamie Heaslip and David Wallace were held up inches short of the line.
But a penalty for not releasing relieved the pressure and the Springboks rubbed further salt into Irish wounds by Steyn slotting an opportunist drop-goal to extend the lead to 10-3 in the 24th minute.
Boks had 'tough season' - De Villiers
By that stage, Paddy Wallace had hobbled off for the Irish to be replaced by Lions centre Gordon D'Arcy.
Sexton maintained his 100 per cent kicking record in international rugby by reducing the South Africans' lead to four points with another superb strike in the 30th minute.
Steyn had two chances before half-time to extend the Springboks' lead but both efforts dropped short in the heavy Croke Park air.
The South African fly-half did not get the distance again five minutes after the break as another opportunity to extend his team's lead was spurned.
Sexton was finding the damp conditions more to his liking and he slotted two more high-pressure penalties in quick succession to have the Irish ahead by the 51st minute.
The Leinster fly-half was finally off-target in the 61st minute as he missed a chance to extend Ireland's advantage.
O'Driscoll 'hungry for more'
Sexton made no mistake four minutes later as he slotted a 50-metre effort and the lead stayed five points as Ruan Pienaar, assuming the kicking duties from Steyn, hit the Irish woodwork.
The Irish fly-half missed a chance to put more than a score between the teams in the 75th minute and as the home pressure continued, Keith Earls looked set to score a try before being bundled into touch.
After being dominated during most of the final quarter, the world champions produced frantic late pressure with Tendai Mtawarira charging to within inches of the Irish line.
However, Brian O'Driscoll's massive hit on Zane Kirchner ensured the vital injury-time turnover as the Irish completed a famous victory.
Ireland: R Kearney; T Bowe, B O'Driscoll, P Wallace, K Earls; J Sexton, T O'Leary; C Healy, J Flannery, J Hayes, D O'Callaghan, P O'Connell; S Ferris, D Wallace, J Heaslip. Replacements: G D'Arcy for P Wallace 34, S O'Brien for Ferris 40. Not used: S Cronin, T Buckley, L Cullen, P Stringer, R O'Gara.
South Africa: Z Kirchner; JP Pietersen, J Fourie, W Olivier, B Habana; M Steyn, F du Preez; T Mtawarira, J Smit (capt), BJ Botha; A Bekker , V Matfield; H Brussow, S Burger, D Rossouw. Replacements: B du Plessis for BJ Botha 48, D Potgieter for Bekker 69, R Pienaar for Steyn 61, J de Villiers for Olivier 65. Not used: CJ van der Linde, F Hougaard, J Deysel.
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