Riyad Mahrez scored the only goal of a hard-fought Group G game at the King Power Stadium five minutes before the interval, the Algeria international flicking home from close range after being picked out by Islam Slimani's header.
However, the Premier League champion was indebted to Schmeichel for keeping Copenhagen at bay in the 90th minute.
The goalkeeper dived to his right to keep out a first-time attempt from Andreas Cornelius, preserving not only the clean sheet but also Leicester's 100 per cent record in the competition.
"We knew it would be tough. Copenhagen tried until the end," Ranieri said.
"It was difficult at the start, but the performance was good.
"It was an amazing save from Kasper Schmeichel at the end. He's in great condition. We deserved the clean sheet tonight."
Leicester - which has won three times in Europe but only twice in the Premier League this season - now leads the pool by five points from Copenhagen and Porto, which recorded a 2-1 victory over Club Brugge in Belgium.
Victory over Copenhagen in the return clash would secure the Foxes' place in the last 16, but Ranieri is refusing to take qualification for the knockout stages for granted despite being in such a commanding position.
"We're in a good position now," he said.
"We want to progress to the knockout stages, ideally in the Champions League but, if not, in the Europa League."