Chile were sent packing after a blunder from Sanchez’s Gunners team-mate David Ospina that gifted Peru a 1-1 draw against Colombia in midweek.
Now Sanchez - who is out of contract at the end of the season - has been challenged to focus all his energies on Arsenal after his international despair.
The 28-year-old has yet to score a league goal this term after the collapse of his £60m move to Manchester City in the summer and his body language has spoken volumes.
He is likely to be left out of the starting line-up for tomorrow’s trip to Watford due to his mental state and having only returned from South America yesterday evening.
But boss Wenger has challenged him to prove he is a true star by bouncing back from his World Cup blow.
“I believe to play at the top level, it is part of dealing with that kind of disappointment,” he said.
“If you want to make a big career it’s a bit of a roller-coaster, because you are sometimes on a high then three days later you’re on a low.
“It’s part of our job. Some deal better than others with it.
“He is in a very difficult mind. Chile just won the Copa America twice, and now they don’t even go to the World Cup.
“Sanchez is 29 at the end of the year, so the next one he will be 33. He expected to go. It’s certainly a big disappointment for him.
“I have no doubt that he wants to perform. He never goes on the football pitch to lose a football game, never.
“Sanchez is a winner. I watched the game with Brazil, what a physical game it was.
“He played against Dani Alves, that was a real battle, believe me.
“You want your players to be in a positive mind, and as well to have positive experiences. To go to the World Cup is the target of every single player who plays football.
“You have always to set targets when you’re a football player. When one is gone, you focus on something else.”
Wenger believes that Sanchez is suffering like David Beckham used to for England.
He said: “I would say you have in every generation a player who carries the pressure of expectation of a nation.
“Why? I don’t really know. But it is like that.
“In France it was [Zinedine] Zidane. In England it was Beckham and we had other players in the team.
“But it was Beckham. Why not Gerrard? I don’t know. But it was Beckham.
“In Chile it’s Sanchez. In Argentina of course it’s Messi. In a national team there is a guy who has to carry the pressure and absorb it and protect a little bit the rest of the team.
“What does that do to that person? It’s someone normally who likes it a little bit. They like to be on the front line.
“Overall what does it mean? It means they have to deal with the lows as well.
“Remember 1998 when Beckham has been sent off after Diego Simeone, he has been slaughtered here, he’s not been picked for the first two qualifiers after for the European qualifiers and he has been treated in this country like a guy who has killed somebody, you know.
“They have to take both the positives and the negatives as well.”
But the Arsenal boss insists Ospina’s error will not damage his relationship with Sanchez.
Peru’s crucial goal came from an indirect free-kick which Ospina touched on its way in.
Had he not, it would have been disallowed and Sanchez would have been on his way to Russia next summer.
Wenger added: “I don’t think that will be a problem, but of course it happens sometimes.”
