Edwards' side still have a mathematical chance of avoiding the drop after fourth-bottom Nottingham Forest's 3-2 defeat against Chelsea just hours after the emotional scenes at the London Stadium.
But the third-bottom Hatters were already coming to terms with their likely return to the Championship due to a vastly inferior goal difference.
It would take a 12-goal swing in Luton's favour on the last day of the season next weekend to save them.
Luton host Fulham, while Forest travel to Burnley, in what is set to be the last game in the top flight for at least a season for Edwards' underdogs.
Accepting the grim reality of their situation, Edwards hugged Luton captain Tom Lockyer in front of their travelling fans as the tears began to flow.
Wales defender Lockyer, who has not played since collapsing on the pitch at Bournemouth after suffering a cardiac arrest in December, was at Edwards' side when the final whistle blew.
"It was really difficult, hugging Locks at the end and seeing how the supporters reacted made me emotional," Edwards said.
"What I said to our players and staff in the dressing room afterwards was 'thank you'. They, along with our fans and the board, have given me the best 18 months of my life.
"I'm really disappointed that I'm the one responsible, I feel that way."
Luton's fate would already have been sealed if Forest had held on to a 2-1 lead against Chelsea at the City Ground.
Late goals from Raheem Sterling and Nicolas Jackson gave Chelsea the points, putting Forest's survival celebrations on ice.
Earlier in the day, Albert Sambi Lokonga headed Luton into an early lead before second-half goals from James Ward-Prowse, Tomas Soucek and George Earthy left the visitors to rue their missed opportunity.
"We've thrown everything at it but just come up short," said Edwards, whose club are in the top tier for the first time since 1991-92.
"It was difficult for us and in the end has just been a bit too much for us. But I've loved it and it fuels the fire to get back into it."
There were emotions of a different kind for David Moyes, who waved farewell after his final home match in charge of West Ham.
Moyes is leaving at the end of the season following a campaign marred by his strained relationship with fans unhappy with his defensive tactics.
"I was certainly emotional because I've been here four-and-a-half years," said the Scot, who won West Ham's first trophy in 43 years in last season's Europa Conference League.
"I was a bit subdued because I knew what the other manager was going through.
"Hopefully I will walk away with people saying something has been done here. Expectations have increased like you can't believe.
"From a team which avoided relegation, now this is a club saying we want to be in Europe every year."
