The match at Parken Stadium in Copenhagen was suspended shortly before half-time after Inter Milan midfielder Eriksen collapsed to the ground with no one around him.
Team-mates and referee Anthony Taylor called for immediate help from medical staff, with the players forming a protective barrier around Eriksen while efforts were made to resuscitate him.
The 29-year-old was carried from the pitch, but positive news followed with the Danish Football Union stating Eriksen was "awake" and set for "further examinations" after being taken to hospital.
Both sets of players decided to resume the game after more than 100 minutes away from the pitch, with Joel Pohjanpalo sealing an historic triumph for Finland shortly before the hour mark against an understandably deflated Denmark side, which missed a late penalty through Pierre-Emile Hojbjerg.
Denmark started on the front foot, with Jonas Wind, Hojbjerg and Eriksen forcing Lukas Hradecky into saves inside the opening 20 minutes.
Thomas Delaney and Martin Braithwaite then fired off target as Denmark continued to push forward in search of an opener.
The game was suspended for almost two hours after Eriksen's collapse, the players ultimately returning to finish the remaining few minutes of the first half before a short half-time interval.
Despite Denmark's dominance, it was Finland that opened the scoring with its first attempt on goal, Pohjanpalo heading through the weak hands of Kasper Schmeichel in the 59th minute.
Kasper Hjulmand's side had a golden opportunity to draw level 16 minutes from full-time when Paulus Arajuuri brought down Yussuf Poulsen in the penalty area, but Hojbjerg's tame spot-kick was kept out by Hradecky.
The reports coming out of Denmark on Eriksen's condition while his team-mates played out the game were encouraging and attention will now turn to his recovery rather than dwelling on an opening-day defeat.
Denmark faces Group B heavyweight Belgium in Copenhagen next Friday (AEST), while Finland takes on Russia in St. Petersburg a day earlier.