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.
Russian Finance Minister Anton Siluanov announced a possible move that Russia can take in response to new US sanctions
When the bill was submitted to Congress on August 2, the reason for imposing the new sanctions on Russia was based on Russia's alleged interference in the US presidential election in 2016, but then something clicked
The Central Bank of Turkey announced measures to protect the financial market of Turkey against the background of the collapse of the Turkish lira and conflict of interests with the United States of America