The number three became the first South African to reach the landmark as he ended on 311 not out with Jacques Kallis also unbeaten on 182 as captain Graeme Smith declared his side's first innings on 637 for two.
A beleaguered England began their second innings after tea, and lost four wickets to leave them reeling with just 102 runs on the board and still trailing by 150 runs going into the final day.
It was another chastening three sessions for Andrew Strauss's side, who have been comprehensively outplayed in every aspect of the game and will be largely unfancied to survive for a draw with South Africa rampant.
England came into the match as the world's top-ranked Test side, but can be usurped by the tourists in the three-Test series, with Smith's side by far the more clinical and efficient, both with the bat, and in the field.
But the day was all about Amla, who continued from his unbeaten century overnight to go on and pass team-mate AB de Villiers's record of 278 to hit the new highest Test score by a South African.
Amla became just the second overseas batsman after Don Bradman (twice) and Bob Simpson to reach the mark in England, in what was a chanceless innings after he was initially dropped on 40 by Strauss earlier in the match.
He shared a monumental 377-run stand with the equally unmovable Kallis, and the pair looked no more likely to be dismissed when their captain intervened with an admirably attacking declaration.
The partnership between Amla and Kallis was the highest third-wicket stand against England by anyone and the first time they had conceded two 250-run stands in the same innings - following the 259 for the second wicket between Amla and Smith.
It was Kallis's 43rd Test century. Only India's Sachin Tendulkar has scored more, with 51, and Kallis celebrated by pointing to his eyes - a show of solidarity for wicketkeeper Mark Boucher who was forced to retire from international cricket after suffering a serious eye injury before the series started.
The innings was also Kallis's first Test ton in England since Old Trafford in 1998, and he was clearly moved by the opportunity to pay tribute to his good friend Boucher, with the injury having shaken him badly earlier on in the tour.
It was another horribly dispiriting day for England, with the hosts having rattled through a staggering 186 overs for a solitary success - that of Smith on day three - after having Alviro Petersen dismissed earlier in the match.
The match was tipped further off balance as South Africa seized control of proceedings once more after tea, removing four England batsmen with sustained aggressive bowling as the hosts' epic stint in the field began to tell with their concentration seemingly shot.
Alastair Cook was the first to go, for a six-ball duck, as he edged a regulation delivery from seamer Vernon Philander behind to de Villiers. It set the tone for the innings, and the visitors already smelt blood.
Jonathan Trott did not fare much better either, as he edged behind off the bowling of paceman Dale Steyn for 10, while captain Strauss looked like a man weighed down by the pressure of the situation.
It was Kevin Pietersen, however, who was the next to go as he survived a few hairy moments at the crease before playing all round a straight, full delivery from Morne Morkel which demolished his woodwork.
Strauss's unhappy spell at the crease soon ended too, as he horribly miscued an attempted pull shot off spinner Imran Tahir and sliced the ball up tamely to Philander at square leg.
Ian Bell (14) and Ravi Bopara (15) were left to guide England to stumps with their wickets unscathed, but it was another hapless session from the hosts in a match in which they have rarely managed to compete.
South Africa celebrated jubilantly as they left the field at the close of play, and Smith's side will have every belief that a victory is in sight on day five. For what the record-breaking Amla and Kallis achieved, it is the very least that the tourists would deserve.