On a wicket that already showed signs of offering significant turn, Dhawan fell for 12 after putting on a second successive half century partnership with Vijay (40) after India captain Virat Kohli won the toss and elected to bat.
Dhawan was taken out by left-arm spinner Dean Elgar after a 68-minute stay.
Vijay, who had made his debut at this ground seven years ago against Australia, was sent back by gangling pace bowler Mornel Morkel in his second spell after the duo had carried India through the first hour of play without being separated.
Earlier, Vijay and Dhawan, who put on an unbeaten stand of 80 at Bengaluru before the Test was washed out, made a steady start against the South Africa pace duo of Morkel and Kagiso Rabada.
Morkel could not extract bounce or pace from the surface while Rabada too failed to trouble the Indian batsmen.
Vijay was hit on his arm guard once by Morkel with a short ball that did not rise before Dhawan glided the bowler to the third man fence with an angled shot for the first boundary of the match in the third over.
Vijay started playing shots and struck Rabada for a couple of fours in successive balls - a lovely flick followed by a glide to square third man.
Seeing his pacers making no impression, Proteas skipper Hashim Amla brought on spin at both ends by introducing Harmer and left-arm tweaker Elgar.
Both the slow bowlers almost immediately started turning the ball, at times sharply, and Vijay - who faced the bulk of the bowling - struggled against Harmer.
He was lucky to get an inside edge when he stepped out to the off-spinner and then smote the same bowler two balls later over the long on region for the first six of the morning.
Dhawan, denied strike for long, was lured out by Elgar soon after 50 was raised in 13.4 overs and he spooned a drive back to the bowler who accepted a return catch to his right.
The left hander's 12 came off 23 balls and was his third low score in the series after bagging a pair in the opening Test at Mohali followed by an unbeaten 45 in Bengaluru.
Pujara started well with two successive fours off leg spinner Imran Tahir's opening over.
But India soon lost in-form Vijay when Morkel trapped the opener bang in front with a full length that moved a shade after pitching. He faced 84 balls and struck three fours and a six.
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