Eriksen collapsed during the Euro 2020 fixture against Finland after suffering a cardiac arrest.
The Inter Milan player was treated with a defibrillator as he was resuscitated by medics before he was stretchered off and taken to hospital.
Kjaer noticing his teammate was in critical condition alerted the medical team immediately he got near Eriksen.
The quick reaction by Kjaer has been described as an "exceptional leadership qualities" in his response to the situation.
Showing gratitude to the warm treatment rendered to him during unfortunate situation, Eriksen stressed;
"You did a fantastic job and saved my life."
"Also a big thanks to my friend and captain Simon and my team-mates in the Danish team for your support, both on 12 June and afterwards.
"Thanks to all the fans who have sent messages to me and my family. It means a lot and has given us strength and support. Thank you."
Kjaer who started the initial CPR procedure before urging his fellow teammates to form human shield around Eriksen as he was attended to by the medical team said;
"It's something that will stay with us for the rest of our lives," said Kjaer.
"The team reacted as a unit…as a team, we made the maximum effort to stick together and try to get through that situation and be as helpful as possible.
"I try to lead the team on and off the pitch, in every aspect of life, as footballers, when we're together. I think it's one of the biggest privileges I have in my career.
"I've known Christian for many years. He has a great family, he has a lot of friends, a lot of team-mates that want the best for him. As long as he's happy and his family is good, I'm happy."
The medical team having done a great job to save Eriksen's life also added;
"We rushed to the field to help [Christian] and to do our job," said Mogens Kreutzfeldt, chief medical officer for Euro 2020 in Copenhagen.
"We did what we should, what we were taught, what we were trained to do.
"Everybody knew their role, everybody knew what to do. We were not emotional at the scene. Afterwards, we were, of course, like everybody. We're very happy and proud of the outcome."
"This year, the President's Award transcends football," said Uefa president Aleksander Ceferin.
"It serves as an important and eternal reminder of just how precious life is and puts everything in our lives into the clearest perspective.
"I would also like to send my very best wishes to Christian Eriksen and his family as he continues his recovery."
