The duke appeared emotional after he was introduced to the stage by wife Meghan, with the pair kissing before he delivered a heartfelt speech.
Harry founded the Invictus Games to aid the rehabilitation of injured or sick military personnel and veterans from across the globe, by giving them the challenge of competing in sporting events similar to the Paralympics.
The Ukraine team received a standing ovation and extended round of applause when they arrived for the ceremony at The Hague’s Zuiderpark on Saturday night.
Meghan, who was invited on stage to speak, said everyone is “standing with” the Ukraine team.
She said: “Good evening everyone. We are so grateful to the Netherlands for welcoming us and for hosting the Invictus Games.
“For each team, my husband and I both recognise it’s been a lot to get here, both physically and emotionally. Not least of which for the Ukraine team, who we are all standing with.”
To huge applause, she added: “Slava Ukraini!”
Harry echoed his wife’s sentiment, saying: “Your bravery in choosing to come and for being here tonight cannot be overstated.
“You told me yesterday when you decided to join us despite all odds, you said you came to be on this global stage, not simply to show your strength but to tell your truth, the truth, of what is happening in your country.
“You know we stand with you. The world is united with you and still you deserve more.
“And my hope is that these events, this event, creates the opportunity in how we as a global community can better show up for you.”
Harry paid tribute to the “boundless humility, the compassion and the friendship” of the Invictus Games.
He said: “I want to personally thank the people of the Netherlands for welcoming us with such warm Dutch hospitality.
“And thank you to the Dutch government, the Ministry of Defence and the Dutch royal family for their support in making this all happen after a two-year wait.”
In her remarks, Meghan said: “One thing I know for sure is that every single moment it has taken to get here will be worth it, because it is here at the Invictus Games that we honour your years of active duty on the field and your continued service to your country, to your family and your community off the field.
“Thank you so much for your service and thank you to all the family and the friends that are here, who’ve been supporting you along the way. Because this is service, this is dedication, and this is the Invictus family.”
The opening ceremony featured live performances by the Kaiser Chiefs, the Johan Willem Friso royal military band, singer-songwriter Anneke van Giersbergen, and rock band DI-RECT.
Princess Margriet of the Netherlands and Dutch Prime Minister Mark Rutte were among those in the audience, with Mr Rutte also addressing the crowd.
Mr Rutte also said he wanted to extend a special welcome to the Ukrainian team.
“No doubt you are here with both mixed feelings and grave concerns about what’s happening at home and some members of your team can’t be here because they are fighting on the frontlines.
“And like you, we are thinking of them.
“And we are also thinking of those Ukrainians in the Invictus community who have paid the ultimate price while fighting in the war.
“They deserve our highest respect. Please know that we are all behind you, that we support Ukraine,” he said.
The opening ceremony came after Harry and Meghan hopped into miniature open-topped Land Rover vehicles driven by two little girls earlier in the day.
The duke also took part in the Jaguar Land Rover driving challenge, sitting in the front passenger seat and helping to navigate a course of bridge structures, ramps and rocks.
Jaguar Land Rover provides financial backing for the games.
The couple arrived at the Zuiderpark around lunchtime and looked casual and relaxed as they chatted to people.
Both sporting sunglasses on a sunny Saturday, the pair met children at a small track next to the main driving course, and each sat in the passenger space next to their young drivers.
Meghan was driven around by five-year-old French girl Mya Poirot, while Harry was driven by four-year-old Dutch girl Scarlet Vroegop.
At one point Meghan appeared to praise Mya for her driving as she could be seen applauding.
The couple made their first public appearance together in Europe since quitting as senior working royals more than two years ago when they attended a reception at the games on Friday.
A crew has joined Harry and Meghan at the games as part of filming for a Netflix documentary called Heart Of Invictus – a series from Archewell Productions following people who are competing at the event.