The hosts were 101 for two at the close.
South Africa were almost halfway to the target with only one wicket down in the last over of the day when wicketkeeper Pant flung himself to his right and held a catch to dismiss Dean Elgar, the home team's captain and star of a similar run chase in the second Test in Johannesburg.
Elgar, on 30, was initially given not out as he shaped to glance Jasprit Bumrah down the leg side. But an Indian review showed a faint spike as the ball went past the bat.
An earlier successful review by Elgar after he was given out leg before to off-spinner Ravichandran Ashwin seemed to anger the Indians.
They clearly felt the ball was hitting the stumps, but the ball tracking technology showed it was just clearing the top of the wicket.
Indian bowling coach Paras Mhambrey said the tourists remained confident that they could win a series in South Africa for the first time.
“There is something in the wicket,” said Mhambrey. “We can create opportunities. There is a patch there where the ball bounces more. We've got to be patient on a pitch like this. We have to hit the right areas and hit those areas more consistently.”
Mhambrey praised Pant, who ran out of partners after hitting six fours and four sixes in 139-ball innings.
“He gave us a great opportunity to win a Test match.”
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