A goal from Carlos Vinicius was enough to ensure that Luton became the first side to lose their opening four games of their first Premier League season since Swindon Town in 1993-94.
"We have created more chances than they have, but they put one over the line and we didn't," said Edwards.
"We did not give them many chances, but the one big one they took, and that is the ruthless nature of this league."
Vinicius' winner leaves Luton at the foot of the table having scored just two goals in four games.
Missed chances cost Luton dearly
The Hatters will have left Craven Cottage wondering what might have been, having spurned several chances to score despite spending long periods absorbing pressure from the home side.
The miss from Morris was followed by Brown's effort against the woodwork, as the summer signing from Stoke hit the post when left unmarked from Issa Kabore's cross from the right.
Luton started the second half promisingly, with Bell shooting wide from Brown's cross, and another fine chance went begging in stoppage time when Lockyer's glancing header flew wide of the target from inside the area.
"We have had three really big chances, we have hit the woodwork with a free header, and we have had a free header right at the death with Tom Lockyer. Amari'i's chance was a very big one as well," said Edwards.
"I also feel we should have had a penalty. Carlton Morris has been fouled when their player is not looking at the ball at the time when the ball is in play. Anywhere else on the pitch, and that is a foul. We have not had a decision there and we have not taken our chances."
Silva's substitutions make the difference
Fulham's last outing came amid a turbulent few days for the Cottagers, with Joao Palhinha's proposed move to Bayern Munich collapsing at the end of the transfer window, before a painful 5-1 defeat by Manchester City.
That loss at Etihad Stadium left manager Marco Silva infuriated by the VAR decision that allowed Nathan Ake's goal to stand, but the Portuguese cut a calmer figure on the occasion of his 100th game in charge at Craven Cottage.
His recall of Palhinha, Fulham's player of the season during the last campaign, gave his side an early edge but they faded and looked short of ideas to combat Luton's resilience as the first half wore on.
But the double substitution, with Harrison Reed and Raul Jimenez making way for Iwobi and Vinicius, proved to be crucial and gave them a cutting edge and creativity. Iwobi was involved in the build-up to the goal before Willian's chipped cross from the left which Luton goalkeeper Thomas Kaminski spilled into the path of Vinicius in the six-yard box.
Despite the single-goal advantage Fulham looked comfortable for much of the rest of the game, holding out to seal the win and their first home clean sheet in the league since beating Nottingham Forest in February.
Palhinha was close to leaving the club before the international break but signed a new contract in the lead-up to this game.
"It is great news for [after] not a great two months with a lot of noise around the club," said Silva.
"It is a clear consequence of last season with players that performed well being linked with other clubs. But I am really pleased for Joao and he showed how he is so important for us, how he is a top player and essential for how we want to play. That is probably a surprise for many, but not for me."