India wrapped up a 113-run victory on Thursday in the first game of the three-match series at Centurion Park, where Kohli continued his run of low scores that has seen him unable to profit from good starts to his innings.
Dravid admitted there had been a lot of "noise" around Kohli in the last few weeks after he was axed as captain of the One-Day International side, causing some controversy back home.
But he said there were no concerns about his form with the bat as India seek a first-ever test series win in South Africa.
"Virat has been phenomenal over the last 20 days we have been here, the way he has trained and connected with the group," Dravid told reporters on Sunday.
"I could not speak more highly about him and the way he has committed to his own preparation and his own practice. On and off the field he has been a fantastic leader.
"We were in a very good space leading into the first test and a lot of that was led by Virat. It is a pleasure to work with someone like him. He batted well but cannot convert those starts."
Dravid believes the runs will come for Kohli, who has not scored a Test century in more than two years.
He has managed 652 runs in his last 14 Tests at an average of 26.08, almost half his career average of 50.34.
"I really feel there is going to be a big run of scores coming in from him," Dravid said. "It might not happen in the next game, though I hope it does, but it will click in place.
"He has been a credit to himself and Indian cricket in the last few weeks."