Federer, who has reached the quarterfinals at Monte Carlo three straight times, set up match point with a serve-and-volley winner and then earned the victory when Lee hit a powerful backhand long.
Lee broke the top-ranked Swiss star in the third game of the first set, but Federer broke back in the fourth and took control after that.
Federer, a 10-time Grand Slam champion, lost last year to Rafael Nadal in the final at both the Monte Carlo Masters and the French Open - the only major Federer has not won.
Federer next plays 12th-seeded David Ferrer of Spain, who has never won a set against Federer in five losses.
Ferrer recovered from 4-1 down in the first set to beat No. 6 Novak Djokovic of Serbia 7-5, 6-4.
The 10th-seeded Czech, whose previous best at Monte Carlo was reaching the second round, will face Robin Soderling in Friday's quarterfinals. The Swede beat Max Mirnyi of Belarus 7-6 (5), 6-2.
Later Thursday, Nadal was trying to extend his winning streak on clay to 64 matches when he faced Kristof Vliegen of Belgium.
Russia has developed an unmanned submarine capable of carrying a powerful nuclear munition. Two years ago, the Americans could not even think Russia could do it
The head of the Russian Finance Ministry, Anton Siluanov, said that the Americans would suffer additional losses if they impose sanctions on Russia's public debt