Fleming made 262 and fellow left-hander Franklin hit 93 not out as New Zealand made 535 for eight at the close of play. The pair put on a record 256 for the eighth wicket before Fleming was bowled by part-time off-spinner Ashwell Prince.
The New Zealanders pressed ahead in bad light before going off when Fleming was out.
Fleming made the highest score by any batsman in a Test at Newlands, while the partnership was the best for any wicket for New Zealand against South Africa. It was also the highest stand for any wicket against South Africa at Newlands.
Franklin, who made his highest Test score, said New Zealand would continue batting Saturday, giving him a chance to make a maiden Test hundred. "Hopefully we can score a few more runs and then it's up to the bowlers to try and do a job on that wicket," he said.
Arthur said: "We've had a quick discussion about batting, how to save the game rather than trying to win it." Arthur said he expected the pitch to continue to play easily.
Both Fleming and Franklin benefited from dropped catches. Fleming gave a straightforward chance to first slip off Dale Steyn five balls after lunch when he was on 136, while Franklin was put down by Jacques Kallis, diving far to his left at second slip, off Makhaya Ntini when he 28.
Two runs later Franklin survived a concerted appeal for a catch behind off Ntini. In the next over he was struck on the helmet by a bouncer from Ntini and fell to the ground before resuming after treatment.
Franklin appeared at a press conference after play with strapping on his left forearm after being hit twice in the same place but said he had suffered bruising only and his bowling would not be affected.
Fleming was missed at slip when he cut a ball hard from South African captain Graeme Smith and Kallis could not get a hand on the ball. He was caught off the shot with which he raised his 250 when he hooked Ntini and Nicky Boje stretched backwards to hold the ball but fell over the boundary.
It was his second six to go along with 31 fours in an innings which lasted 576 minutes. He faced 423 balls.
It was a masterly performance by Fleming. He acknowledged after reaching his century on the first day that he did not have a good record of converting fifties into hundreds, but he has now converted three out of nine centuries into doubles.
He fell 12 short of his career-best 274 not out against Sri Lanka in Colombo in 2002/03.
Fleming went past the previous highest score by a New Zealander against South Africa, 170 by Scott Styris in Auckland in 2003/04, and went on to notch the highest score by a visiting player in 40 Tests at Newlands, beating the 189 by Jim Burke for Australia in 1957/58 before passing Herschelle Gibbs' ground record of 228 for South Africa against Pakistan in 2002/03.
Franklin, who has made a first-class double century, confirmed his reputation as a capable batsman as he and Fleming humbled the South African bowlers on an easy-paced pitch, making a mockery of Smith's decision to send New Zealand in.
By the close Franklin had batted for 299 minutes, faced 220 balls and hit ten fours.
Franklin admitted the pitch had not helped the bowlers but said he was confident that his team could press for a series-levelling win.
"There are still three days to play. We've got a reasonable seam attack and a world-class spinner in Daniel Vettori, while Jeetan Patel is making his debut and he's a very good off-spinner."
The only wicket to fall before Fleming was that of Daniel Vettori, in the third over of the day and the first with the second new ball. Vettori pulled Ntini's first delivery for four but in trying to repeat the shot two balls later he spooned a catch to Andre Nel at mid-on.
Instead of whipping through the rest of the batting, South Africa's bowlers failed to find a consistent line and length and did not manage to bowl a maiden until Kallis sent down the 36th over of the day.
By then New Zealand were in command. There were only three more maidens in the rest of the day.
The start of play was again affected by the vagaries of Cape Town's autumn weather. After a delay on the first day because of dew, a portable hovercraft was brought in to dry the outfield, but fog swept over the ground and play only started 55 minutes before lunch, with 83 overs scheduled for the day.
South Africa lead the three-match series 1-0.
New Zealand 1st innings: