Smith held four catches on Sunday, taking his career total to 223, as South Africa were bowled out for 284 in reply to Australia's first innings of 390.
Nic Maddinson, 17 not out, and Travis Head, 10 not out, added 33 runs off just 5.4 overs to stretch Australia's overall lead to 139 before bad light ended play for the third day in a row. Maddinson, who laboured to a 33-ball duck in the first innings, hit two fours off Marco Jansen in the first over of the second innings and added a six off Wiaan Mulder.
Australia's bowlers applied relentless pressure on an easy-paced pitch. Six Protea batters reached 30, but David Bedingham's 49 was the top score. Captain Pat Cummins took three for 54, while Mitchell Starc, Josh Hazlewood and Nathan Lyon claimed two wickets each.
Smith started the day on 219 catches, one behind Joe Root of England. He equalled the record when he caught Temba Bavuma, and broke it when he lunged to his left from second slip to hold a sharp edge from Bedingham off Cummins. He added a spectacular catch to dismiss Verreynne, holding the reverse sweep in an outstretched right hand, and finished with an edge from last man Anrich Nortje off the bowling of Beau Webster.
Smith's career haul of 223 catches exceeds the total of all but 11 specialist wicketkeepers.