The anniversary on 25 April marks the first major World War One battle involving Australia and New Zealand.
Monsoons in Darwin had said the prize of A$1000 (Â£600; $750) would be donated to military charity Solider On.
But veterans organisation the Returned and Services League (RSL) described the event as "incredibly inappropriate".
After public anger, the bar said in a statement: "We apologise for our naive approach and lack of proper thought when organising our activities calendar."
In Australia, it is against the law to use the term Anzac to promote businesses and events without permission from the Department of Veterans' Affairs. Companies can be fined up to A$51,000 while individuals face up to 12 months in jail.
It was not the only accusation of disrespect on the important national occasion
A 32-year-old man was arrested and charged with offensive behaviour after shouting out anti-war comments during the one-minute's silence at a dawn service in Sydney.
TV presenter Yassmin Abdel-Magied apologised after a backlash against her Facebook post suggesting the day should be spent thinking about asylum seekers.
A string of venues in Sydney's famous LGBT entertainment district Oxford Street were also criticised for advertising events featuring scantily clad models in military clothing.
Meanwhile, a terror warning two weeks ago has been blamed for a low turnout at Gallipoli.
The site in Turkey where Anzac troops were engaged in conflict has become a place of pilgrimage for many Australians and News Zealanders.
A record 10,000 people attended the ceremony for the 2015 centenary, but this year only about 1,000 visitors attended, and were outnumbered by security.
An Australian bar has cancelled plans to hold a wet T-shirt competition to mark Anzac Day, saying it had been a "naive" plan.