After a goalless first half in which Denmark looked lively and had more possession, Australia scored on the counter in the 60th minute when Leckie got past the Danish defense and twisted into space before slotting the ball home.
Australia finished second with six points, level with group winners France after the world champions lost 1-0 to Tunisia. Denmark were bottom on one point with Tunisia third on four.
Graham Arnold's side next face the winners of Group C, which was led by Poland after two rounds. The Poles face Argentina later on Wednesday in their final group match.
The Australians have qualified for the knockout rounds for only the second time, having also done so 16 years ago when they lost 1-0 to Italy in the last 16.
Denmark created chances in the first half and could have taken an early lead but Jesper Lindstrom's shot from close range was blocked before Australia goalkeeper Mat Ryan parried away an attempt from Mathias Jensen.
The Australian defence was guilty of gifting space to Joakim Maehle on the left in the 19th minute as the defender drilled the ball into the box, but Ryan once again came to Australia's rescue, kicking the ball away.
Australia began creating chances and threatening the Danish defence as forward Duke Mitchell took aim from range, while Riley McGree had a shot saved by Kasper Schmeichel.
They finally broke the deadlock at the hour mark as Leckie timed his run perfectly and sent the ball through Maehle's legs before beating Schmeichel.
Denmark thought they had been handed a lifeline 10 minutes later when the referee awarded a penalty for a foul on Kasper Dolberg, but it was quickly overturned after the flag was raised for offside.
MNA/PR