Madigan stepped up to the plate with eight points off the tee, and the Leinster man could not hide his emotion on the final whistle, shedding tears as Ireland dug out the win.
And with Sexton's fitness now in doubt for Ireland's quarter-final clash with Argentina, Schmidt is confident Madigan can cope with being the main man for his country.
"You're always delighted when young guys come through the system," he said. "I think if anyone saw Ian away from the game they would see how hard he works.
"He's incredibly hard-working, his goal-kicking percentages are very, very consistent, and that gives us confidence.
"He has the courage to make sure that defensively we didn't feel that whatever they threw at him, he wouldn't be shy away from.
"I thought he was really assured for us. When he needed to carry, he carried, when he needed to link up, he did.
"One thing that helped Ian was the quality of the forward ball that Jamie [Heaslip] and the pack offered up to him.
"I think it allowed him to get on the front foot and that always makes a ten's job that little bit easier."
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