Pochettino was delighted with Rodriguez's first Saints' performance as a full England international.
He said: "That was his first game with the national side, that's never easy.
"Maybe his confidence dropped a little bit, but not just Jay, the whole English national team didn't play that well against Chile.
"It's clear that with Southampton he feels a lot more confidence, he feels a lot more at home, a lot more at peace.
"As with any player, we need to give him the confidence, we need to give him the encouragement to keep on playing well, if they're playing with their club or if they're playing with their national side."
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