A 1-0 defeat to FC Midtjylland in Denmark on Thursday night was enough to send Saints packing out of the Europa League at the play-off stage, with former Celtic striker Morten Rasmussen scoring the only goal of the game.
Southampton had a number of chances in the early stages but could not get an all-important away goal, with Koeman infuriated after the game as he felt his side were denied a clear second-half penalty.
That followed the Dutchman's angst after a 1-1 draw at St Mary's in the first leg where Steven Caulker saw a goal ruled out for a foul in the build-up.
Koeman said he was left disappointed by the officials, whom he felt missed 'easy' decisions across the tie and focused on them rather than criticising his own players.
But his captain did not toe the party line and Fonte, who saw an early diving header cleared off the line, was quick to hold his hands up when asked for his reaction to Southampton's exit.
"We weren't at our best and we weren't good enough," he said.
"Credit to them, they stuck to their game plan and executed it well
We have ourselves to blame
We need to show our face, accept the mistakes and understand that there is another day tomorrow.
"I feel sorry for the fans, but no-one is more disappointed than us
We worked hard last season to be in the Europa League and to be playing European football, but it didn't happen.
"I thought that we had more quality than them - and on a couple of different nights we could have easily beaten them, but the fact is that we didn't so we can only blame ourselves for that."
Southampton will now refocus on their stuttering domestic form having taken two points from their opening three Premier League fixtures.
Newly-promoted Norwich are the visitors to the south coast on Sunday and Fonte knows the Canaries will be hoping to take advantage of any European hangover and urged the fans to stay with the team.
"We have a big challenge ahead on Sunday in the league," he added.
"The Premier League is hard and we need to move on
That's our focus now.
"We need to pick ourselves up
Obviously it's hard, with this disappointment, to move on, but there's no alternative.
"When we go back to the training ground, it's time to focus completely on getting the three points against Norwich
I just want to ask the supporters to be behind us like they always are because we need them more than ever.
"We're lacking a little bit of confidence so we need a little bit of a boost on Sunday from the St Mary's crowd to make it a fortress like it was last year."
One positive for Koeman ahead of the visit of Norwich is the hope that both Victor Wanyama and Sadio Mane will be fit to play some part after both missed the tie in Denmark.
Source : PA