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.
The discovery of the submarine has unveiled a few "inconsistencies." For example, how can one explain the fact that the sub was found where it needed to be searched for from the start?
When on a state visit to Singapore, Russian President Vladimir Putin promised to revisit the discussion of the 1956 Declaration between the USSR and Japan regarding the issue of the peace treaty with Japan
The TurkStream, which runs along the bottom of the Black Sea from Russia's Anapa to Turkey, will consist of two lines, each with a capacity of 15.75 billion cubic meters of gas a year