The 29-year-old midfielder fell to the ground towards the end of the first half of the game at the Parken Stadium in the Danish capital, with English referee Anthony Taylor calling medical staff onto the pitch, who administered CPR.
UEFA confirmed the Group B match had been suspended due to a medical emergency, and the Inter Milan player was later taken to a nearby hospital.
Eriksen's partner Sabrina and his father Thomas are with him, and the player is understood to have spoken to his team-mates from the hospital.
The Danish football association issued a statement shortly after 7pm UK time reading: “Christian Eriksen is awake and his condition remains stable.
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"He remains hospitalised at Rigshospitalet for further examination.”
The match later restarted amid claims the players had asked to complete the fixture.
UEFA President Aleksander Čeferin said in a statement: “Moments like this put everything in life into perspective. I wish Christian a full and speedy recovery and pray his family has strength and faith.
"At these times, the unity of the football family is so strong and he and his family carry with them the good wishes and prayers of everyone.
"I heard of fans of both teams chanting his name. Football is beautiful and Christian plays it beautifully.”
Messages of support
Eriksen's former club Tottenham Hotspur tweeted: "All of our thoughts are with Christian Eriksen and his family,” while countless other teams and football personalities posted messages of support as medical teams tended to the player on the pitch.
Fabrice Muamba, the former Bolton midfielder who collapsed after suffering a cardiac arrest during a televised FA Cup game in 2012, tweeted "Please God" following the incident.
The Scottish national team sent a message of support from its official Twitter account reading: “The thoughts of all at the Scotland National Team this evening are with Christian Eriksen, his family and @DBUfodbold.”
Scotland’s Group D opponents England cancelled a press conference previewing their Euro 2020 clash against Croatia following Eriksen’s collapse.
Manager Gareth Southgate and captain Harry Kane who was Eriksen’s team-mate at Spurs, were due to preview the Group D contest at 6.30pm UK time on Saturday, before the distressing events in Copenhagen began to filter through.
Anger at BBC
The BBC later issued a statement apologising to viewers following a number of complaints that coverage had continued while Eriksen received treatment.
Gary Lineker, who was presenting coverage of the match and was clearly in shock along with pundits Cesc Fabregas, Alex Scott, and Micah Richards, wrote on social media: “I understand some of you would have been upset with some of the images shown – we were too. Obviously these were the host pictures and out of our control.
"They should have stayed on a wide of the stadium. Apologies.”
A spokesperson for the BBC added: "Everyone at the BBC is hoping Christian Eriksen makes a full recovery.
"We apologise to anyone who was upset by the images broadcast. In-stadium coverage is controlled by UEFA as the host broadcaster, and as soon as the match was suspended, we took our coverage off air as quickly as possible."