Facing a break point at 4-4 in the first set, Henman was helped by a controversial over-rule from the umpire against ninth seed Lee, before the Brit won the set 6-4.
The 10th seeded Henman eventually won the second set in a 7-5 tie break after games went with serve.
Earlier, Federer easily accounted for Germany's Benjamin Becker 6-3 6-4 to reach the final of his maiden tournament in Japan.
The Swiss player took just one hour to dispense with the 14th seeded Becker, who was the last man to play Andre Agassi, and was confident going into the final.
"It was pretty straightforward," Federer told reporters.
"The feeling's good. My dream was to be in the final.
"The last little bit is a bonus. It's a great opportunity to win maybe my first tournament in Japan."
Federer struggled in his quarter-final against 1,078th-ranked Japanese player Takao Suzuki on Friday and was briefly in danger of suffering the worst defeat by a world number one in ATP Tour history, when he was one set down.
"The big difference was I got a break behind early (against Suzuki), so once I got a break, I was in control and make sure I can hit a lot of first serves in, because that is the big factor if you win or lose a point here," said Federer.
"Because if you make your first serves, the chance is so big to win the game comfortably. That's what I tried to do," he added.
In the women's final, local favourite Aiko Nakamura will play French top seed Marion Bartoli, after the Japanese player defeated Chan Yung-jan of Taiwan 7-6, 2-6, 6-4, and Bartoli thrashed compatriot Camille Pin 6-0, 6-1.