He confirmed the chosen city in a video message, which ended with the backing of pop star Kylie Minogue.
The games are a Paralympic-style multi-sport event ranging from archery and wheelchair rugby to road cycling and swimming.
The event began in London in 2014 and visited Orlando, Florida, in May ahead of Toronto next year.
"We have raised the bar higher with each successive games. I am happy to say the Invictus Games journey won't end there," said Harry.
He said the competition had "shown us what can be achieved when wounded, injured and sick servicemen and women rediscover their fighting spirit through sport".
Prince Harry has been heavily involved in the organisation of the event, after attending the Warrior Games, a similar gathering held in the US.
He himself served 10 years in the Armed Forces and saw action in Afghanistan twice.
The Invictus Games
The announcement video, filmed at the Tourism Australia offices in Australia House, London, shows Harry's mobile phone ringing to the sound of the Kylie Minogue hit I Should Be So Lucky.
The prince, surrounded by members of the Australian military, announces to cheers that the caller is Kylie.
He says: "Hi Kylie, good timing, the 2018 Invictus Games is coming to Sydney, can I guarantee the Aussies are going to bring it?"
In her own video message in reply, she says: "Hey Prince Harry, listen - we're stoked the Invictus Games is coming to Sydney, but you don't need to tell the Aussies to bring it - it's guaranteed, no worries."
The prince, who is patron of the Invictus Games Foundation which oversees the delivery of the tournament, said Sydney was chosen because it is an iconic location with a proud military heritage and a population that is "absolutely sports mad".
The 2018 competition is set to take place in New South Wales from 18-29 October.
Prince Harry has named Sydney as host of the 2018 Invictus Games for injured, wounded and sick armed forces veterans.