Spurs were favourites to reach the quarterfinals after securing a 2-0 lead from the first leg in London but Zagreb stormed back with a sensational hat trick from Mislav Orsic to claim a 3-2 aggregate victory.
Manager Jose Mourinho accused his players of lacking professionalism after the defeat and Lloris echoed his comments.
"I think we are more than disappointed, it is a disgrace," Lloris said. "I hope everyone in the changing room feels responsible.
"The taste of defeat is more than painful. We are a club full of ambition, I just think at the moment it is a reflection of what is going on in the club.
"We have a lack of basics, fundamentals, our performances are just in relation of that. Mentally we should be stronger, more competitive."
"When you are not ready at this level, you pay," he added. "There is quality everywhere and if you don't respect the opponent they can punish you, that's what happened."
Tottenham, who are eighth in the Premier League, visit Aston Villa on Sunday.