South Africa closed on 403 for two for a first innings lead of 18 runs at The Oval.
Smith stroked 131 before he played on, via his pad, to Tim Bresnan after lunch but Amla continued the tourists’ dominance supported by Jacques Kallis, who was 82 not out at the close.
It was Smith’s 25th test century and he became the seventh player to achieve the feat of scoring a hundred in his hundredth test after Colin Cowdrey, Gordon Greenidge, Javed Miandad, Alec Stewart, Inzamam-ul Haq and Ricky Ponting.
Amla notched his 15th test century while Kallis is eyeing an incredible 43rd.
England are seeking to hold on to their top ranking while the Proteas can leapfrog them if they win the three-match series.
On current form, the tourists have an excellent chance to go 1-0 up, but a placid batting surface could make it difficult to dismiss England a second time, cheaply.
Smith and Amla cashed in while the sun shone and England no doubt rued the dropped chance that skipper Andrew Strauss made at first slip the previous evening when Amla was on 40.
Amla has otherwise been faultless and the ease with which his runs came said much about his ability to penetrate the field against quality bowling as much as the conditions.
His innings has so far spanned 369 balls in nine hours. But, because of the occasion, Smith’s six-hour innings will be remembered more, while he shared in a potentially match-defining partnership of 259 with the right-handed Amla.
Smith reached a special century with a back cut for four to the third man boundary off Bresnan shortly before lunch.
His innings was a tale of two 50s. His first was the slowest of his career, reaching the milestone in 160 painstaking deliveries as he was especially tested by off-spinner Graeme Swann on a turning pitch.
He faced just 41 deliveries to go from 50-100, however, eventually turning the pressure back on to Swann, who conceded 21 runs off the final two overs of his first spell.
Kallis, the fourth-highest run scorer in tests, was probably not the sight England’s fielders wanted to see at 260 for two and he duly added further misery to England’s day with an innings of authority that included ten boundaries.
AB de Villiers, Jacques Rudolph and JP Duminy are still to bat.