A 2-2 draw in the final round of qualifying games means that Gordon Strachan’s side miss out on a play-off place to Slovakia on goal difference, extending their absence from major tournaments beyond the 20-year mark.
Celtic striker Griffiths had given the Scots a half-time lead but Roman Bezjak’s double put Slovenia in front before Robert Snodgrass equalised late on.
They could not find a winner, though, and Griffiths told Sky Sports: “We put so much into qualifying, in the last four or five games we have been outstanding and we just needed that one final push.
“The goals we conceded were sloppy and then we get the equaliser and then we should have got one at the end.
“It was always going to happen at some point, we were going to be put under pressure, they are the home team, they were going to come forward and they wanted to win the game.
“It’s disappointing, but we can hold our heads up high, for the last four or five games we’ve been great.
“(The manager) couldn’t ask for any more, we have played 90 minutes twice in a matter of days and to still keep running at the end to try and prove people wrong is a credit to the boys.”