The Tigers, European champions in 2001 and 2002, have become England's first major casualty of a tournament Guinness Premiership clubs look like dominating this season.
London Irish, Gloucester, Saracens and holders Wasps are all queueing up to claim prized quarter-final places.
Leicester, though, must reflect on just a third pool stage exit in 11 European campaigns after being undone 17-12 by Andy Robinson's Edinburgh at Murrayfield.
Loffreda, who led Argentina to the World Cup semi-finals just three months ago, said: "Obviously, we are bitterly disappointed. We had three penalties in the last 10 minutes, yet didn't go for goal with a single one.
"This was a game we could have won, and if we had kicked a couple of goals in that period we could have put pressure on Edinburgh and the result might have been very different.
"But we were not surprised by Edinburgh - we expected them to be strong at home."
Tries by Ross Ford and Phil Godman saw Edinburgh avenge a 39-0 Pool Six drubbing at Leicester earlier in the tournament, and former England head coach Robinson said: "It was one of the best performances since I came to the club.
"And the fact it was against one of the strongest teams in Europe made it all the more special."
Toulouse, despite a 20-13 loss to Leinster in Dublin, will top the group if they subdue Edinburgh at home next Saturday.