The duke thanked Australia for the welcome he and his pregnant wife Meghan had received in the country, as they take part in a 16-day tour which will also take in New Zealand, Fiji and Tonga.
He said: “I have been so proud to be able to introduce my wife to you and we have been so happy to be able to celebrate the personal joy of our newest addition with you all.”
Athletes from 18 nations will take part in 13 sports in the Australian city over the next week as part of the Games for injured service personnel.
Harry said: “if we’re being honest, we know that, in many nations, the place of our men and women in uniform became too often undervalued in the decades after the Second World War.
“The sacrifice and character of our troops never changed, but society’s recognition of them too often wavered.
“But that has changed. A new generation – the Invictus Generation – is defining what it means to serve. And we are all taking notice.
“The Invictus Generation has chosen to serve their countries in conflicts that are complex and dangerous and far too often this dedication goes unrecognised.
“They have reminded us all what selfless duty really looks like.”
The Duke was pictured making his final preparations for the ceremony earlier on Saturday, with his wife the sole audience member.
Harry added: “With the help of medical science, the Invictus Generation has not only survived injuries that in past conflicts would have been fatal, but has also chosen to fight back from the darkest of places to be here tonight.
“They have shown us the true meaning of resilience.
“When they have been open about their hidden emotional and mental wounds, the Invictus Generation has shown us that, in today’s world, being tough means being honest about how we feel – both inside and out.
“When they have taken to the tracks, fields and pools in London, Orlando, and Toronto, the Invictus Generation has exemplified sportsmanship, bravery and world class athleticism.
“They have shown us all that the most difficult challenges can be overcome.”
The opening ceremony, which included a parade of nations, performances from the Australian Military Wives Choir and Birds Of Tokyo, took place 45 years to the day since Harry’s grandmother, the Queen, opened the Sydney Opera House.
At a reception in the building’s Bennelong Restaurant, the duke expressed concern about the weather and the potential impact on the ceremony.
But the heavy rain came to a sudden end shortly before the couple were escorted into the venue by Premier of New South Wales Gladys Berejiklian.
“I was a bit worried,” Harry told Australian Prime Minister Scott Morrison as they walked across the room. “Thankfully it looks like the sky is clearing now.”
Meghan, who was wearing a Stella McCartney dress and coat by X-Files actress Gillian Anderson for Winser London, smiled as the premier told the couple: “There a lot of people looking forward to meeting you.”
At the opening ceremony, Mr Morrison spoke of the different definitions of “winning” at the Games.
He said: “Those who are already here know victory in the toughest arena there is.
“In the isolation and darkness of the night, where doubt and hopelessness surround, and when they look deep into themselves and answer the phone that torments them at this time with the cry ‘I will not be overcome’.
“So what does winning look like at this Invictus? It will be different for every athlete.
“Gary Robinson, a survivor of a Black Hawk crash, said winning for him is ‘seeing my family, friends and supporters in the stands’.
“Sonia Newman, a veteran who had to learn to walk again, said ‘it’s showing my kids what I can achieve’.
“And Matthew Brumby, a wheelchair rugby competitor, said winning is ‘to get out and have a crack’.”
Harry and Meghan are expected to watch competitors take part in the road cycling and sailing on Sunday, before their tour continues on Fraser Island in Queensland.