Pakistan were 213 for one in their follow-on innings at close of play, still 208 runs short of making South Africa bat again.
Pakistan captain Masood made 102 not out, his sixth Test century, and Babar scored 81, his third successive half-century in the series.
Babar, who has not scored an international century since August 2023, fell short again when he was caught in the gully off Marco Jansen 14 minutes before the close of play.
He stood in disbelief in the first innings after glancing a loose ball from 18-year-old debutant Kwena Maphaka down the leg side to be caught by wicketkeeper Kyle Verreynne for 58. He was similarly reluctant to go in the second innings.
"Both innings I batted well, then unfortunately I made a mistake," said Babar. "It was very disappointing because I started well but did not finish well. Unfortunately as a batsman you make a lot of mistakes."
The Masood-Babar partnership of 205 was the highest for the first wicket for Pakistan in Tests against South Africa. It also exceeded Pakistan's first innings total of 194, which prompted South African captain Temba Bavuma to enforce the follow on.
Babar insisted there was no change in mindset between the two innings.
"The coach (Aaqib Javed) said 'just play your game'. Unfortunately we did not do it in the first innings but in the second innings me and Shan started well and God willing tomorrow we can do the same thing."
A tale of two innings
It was the second significant partnership of the day for Babar, who was forced to open the batting after Saim Ayub was ruled out of the match with a broken right ankle.
Babar and Mohammad Rizwan (46) put on 98 for the fourth wicket in the first innings before Babar's dismissal triggered a collapse.
South African bowling coach Piet Botha said the South African bowlers needed to stay with the "discipline and patience" that had brought them six successive wins and a place in the World Test championship final.
"We did well to bowl them out in the first innings but it's Test cricket. When a partnership develops nothing changes. The wicket was at its best for batting but we had to attack, keeping slips in place. Tomorrow is a new day."
Masood and Babar showed in the second innings that there were no demons in a pitch on which South Africa scored 615.
They kept the scoreboard moving with positive stroke play and good running between the wickets and seldom looked in danger of getting out.
South Africa's frustration boiled over when Wiaan Mulder, after bowling a ball which Babar struck back firmly, threw the ball at the batsman, who was on 56.
The incident led to an exchange of words, which also involved wicketkeeper Verreynne, and an intervention by the umpires.
"It was in the heat of the moment, it is sport," Babar said with a smile when asked about the incident.
