Marcus Smith marshalled England's backline well from fly-half before landing the match-winning kick.
England scored 27.
Tries: Tuilagi, Steward, and Quirke Cons: Smith 3 Pen.
South Africa has 12.
Try Mapimpi Pen: Pollard, Jantjies.
England signed off in style as they outsmarted and outfought South Africa in front of a rapturous Twickenham crowd.
The hosts had jumped out to a 17-6 lead after the first half.
The boot of Handre Pollard kept the Boks in the fight after the break.
The world champion's mix of rock-solid mentality and physical strength seemed to have won it when Makazole Mapimpi went over for a try on 69 minutes.
Eddie Jones predicted that England would have to fight to the end, and Marcus Smith was able to slot the winning penalty in the 78th minute.
The head coach's tactical reboot has been vindicated by the victory over the world number ones and world champion.
It felt like a landmark win as Smith finally kicked the ball dead after Andrew Brace had to rule on an aerial contest in the shadow of England's posts.
The promise of what might be in France in 2023 is one that points the way away from the defeat in Yokohama in 2019.
England defeated the best.
South Africa captain Siya Kolisi and England replacement prop Will Stuart were both sent to the bin in the second half.
England had a performance that buzzed with energy, invention and no confrontation in the first half.
The first scrum skirmish ended with England prop Kyle Sinckler grinning widely at his opposite number Ox Nche and the referee's arm aloft to indicate a penalty to the hosts.
Jamie Blamire's first line-out as England's starting hooker was a clever trick play, bulleted to a grounded Maro Itoje inside the Springbok 22m.
The breakthrough try came shortly after as Smith and centre Henry Slade's quicksilver hands beat the Boks' attack and gave Tuilagi a run into the corner that the covering Pollard was never going to stop.
The Lions series victory was still fresh in the memory when South Africa arrived at the venue. They have looked vulnerable to a backline that can outflank their defence.
Australia beat them twice in a week in September with the canny Quade Cooper at the wheel, while Finn Russell's eye for space opened them up against Scotland at Murrayfield last weekend.
Even when England's front row were crumpled at a scrum on halfway, Blamire's hook and Ben Youngs' smart pass moved the ball away fast to create the hosts' second try.
South Africa was covered back without fully recovering, and Steward was able to punch over from short range.
England's brave new world seemed almost unbelievable when Smith kicked a penalty after 24 minutes. The opposition was on the ropes and the hosts were crowing.
It was going to be hard for a while.
Pollard's boot, although not perfect, chipped away at the lead, and the introduction of Malcolm Marx and Steven Kitschoff, the fabled 'Bomb Squad' front row replacements, seemed to sap England's belief.
South Africa was two points behind when Pollard kicked another penalty with 25 minutes remaining.
South Africa came on strong after England's scrum was stuck in reverse.
In such situations, teams are bonding and names are made.
England's young stars stood up. The South Africa line was held up by a driven maul and a superb supporting line before the South Africa line was taken by Joe Marchant's break.
The hosts landed the final punch after Mapimpi's score and Frans Steyn's penalty seemed to have squeezed England out of the contest.
Smith kept his cool when South Africa was on the floor at the end of the game and he scored with 15 seconds left on the clock.
