Van der Burgh, the world record holder, was second fastest among the qualifiers for the 50m breaststroke semifinals.
He touched the wall in 26.27 seconds, 0.20 behind British swimmer Adam Peaty, while fellow South African Giulio Zorzi also progressed beyond the heats, clocking 26.84 as the 13th quickest qualifier for the penultimate round.
Meanwhile, the SA mixed 4x50m freestyle relay team were ranked seventh in the opening round of their event in 1:33.28, booking their place in the final.
Leith Shankland, who clocked 47.72 to progress as the 13th fastest swimmer in the 100m freestyle, and Sebastien Rousseau, who touched the wall in 53.55 to qualify 15th in the 100m individual medley, booked their places in the semifinals of their events.
Chad le Clos, who was the quickest of the 50m butterfly semifinalists on Friday, was also set to compete in the evening session, in search of his third gold medal at the global championships.