Harry said he hopes the games, which are taking place in The Hague in the Netherlands, will create the opportunity for the world to "better show up" for the country of Ukraine.
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."
