Denmark created the first chance of the match after three minutes when Fredrik Borsting got down the right-hand side to the byline and cut the ball back to Emil Larsen, who dragged his shot wide.
That was an early warning side for South Africa, but it wasn't heeded as Denmark created further opportunities, targeting Abbubaker Mobara at right-back as they played through-balls between him and the centre-back. Twice Mobara did not track his runner but Khune saved both efforts, one of them at point-black range with his head.
The South African defenders were also let off the hook when Denmark floated a routine ball into the box and two players were allowed to run free, unmarked, with a poor offside trap being deployed. Fortunately, the chance was wasted.
South Africa's best chance came as early as the seventh minute when Keagan Dolly, pulling the strings just behind the attacking three, combined cleverly with Gift Motupa on the edge of the area. Dolly received the one-two and then got into the box before putting it on a plate for Motupa, whose close-range shot was saved by Denmark goalkeeper Jeppe Hojbjerg.
The young Bafana Bafana also hit the woodwork twice, first when Lebogang Mothiba's follow-up shot was tipped onto the crossbar by Hojbjerg after Dolly ran onto Rivaldo Coetzee's pass over the top, then when Dolly took aim from 35 yards out and crashed a stunning shot onto the same part of the crossbar, again via a slight touch from Hojbjerg.
Owen da Gama's side bossed possession at the start of the second half without creating many openings before Denmark took the lead in the 69th minute. Substitutes Jacon Bruun Larsen and Robert Skov ran through the middle, unnopposed by either Erick Mathoho or Rivaldo Coetzee, who had played them onside, before Bruun Larsen cut the ball back to Skov to sweep home.
South Africa tried their best to get back into the match – Da Gama had brought on Menzi Masuku for Tebogo Moerane at half-time and Tashreeq Morris for Lebogang Mothiba in the 61st minute – but their best opportunities were two Motupa shots that were kept out by the impressive Hojbjerg.
Da Gama then brought on Phumlani Ntshangase for the ineffective Kwanda Mngonyama with about 15 minutes remaining, but the damage had already been done, and to make matters worse, Erick Mathoho was stretchered off with what looked like a knee injury, and Mobara also rode off injured on the back of Khune in the closing stages.
Denmark now top the group with three points, with Brazil expected to beat Iraq in the next game.
South Africa U-23: Khune, Mobara, Mathoho, Coetzee, Moerane (Masuku 46'), Mngonyama (Ntshangase 77'), Mekoa, Modiba, Dolly, Motupa, Mothiba (Morris 61')
Denmark U-23: Hojbjerg, Puggaard, Gomes, Mathiasen, Maxso, Vibe, Brock-Madsen (Bruun Larsen 64'), Borsting (Laursen 84'), Larsen (Skov 61'), Gregor, Jonsson