Hibernian manager Terry Butcher and his Hearts counterpart Gary Locke paid tribute after the teenagers died within hours of each other.
Jamie Skinner, 13, was playing for Edinburgh-based Tynecastle FC against Spartans FC - after leaving Hearts just weeks ago -when he collapsed during the match in the Saughton area of the Scottish capital on Sunday.
Hibernian player David Paul, 18, died in his sleep in the early hours of that morning.
Butcher said the club was in a state of disbelief after David's death.
He said: "Every single member of the squad, the club, everyone at Easter Road and the training centre, are absolutely devastated. It is beyond everyone's comprehension.
"He had been at Hibs since he was 10 years old, he was in the second year as a full-time player and he was much loved by everyone.
"He was a quite unassuming boy who got on with his job, a great professional, a great member of the squad."
David's death comes five years after his older sister Jennifer died in her sleep, aged 16.
Butcher said: "I have three hulking boys and I just don't know what would happen if I lost one of them, so our thoughts are with his family. It is terrible news for them.
"Both clubs in Edinburgh are mourning the loss of good characters as well as good players. It really puts it into perspective.
"All we do is play a game of football. Period. The things that really matter in life are family and life itself."
He said players struggled as they trained for their next match on Boxing Day against Ross County in Dingwall.
"We know how much David would have liked us to carry on and how much he would have wanted Hibs to win. So we will certainly play with an extra feeling of wanting to do well for David and his family and cherish his memory in the right way, by going out on Thursday and winning."
Speaking at his pre-match press conference ahead of Hearts' game against Kilmarnock on Boxing Day, Locke said the deaths of both boys were "tragic".
"We are all devastated," he said. "It puts things into perspective. We think we are having difficult times but for us and Hibs, to lose two young players is tragic.
"My thoughts go out to both players' families because it must be absolutely horrific for them.
"He (Jamie Skinner) turned up expecting to play football and he has lost his life so deepest sympathies for both families. Everyone at the football club is hurting."
All Tynecastle FC teams will hold a minute's silence before their next game.
A spokesman for the Scottish Professional Football League said: "We all offer our sincere condolences, thoughts and prayers to the families, friends and team-mates of David Paul and Jamie Skinner."
There are no suspicious circumstances and post-mortems are to be held this week.