The 17-year-old lightweight, hoping to become the youngest Olympic boxing champion since American Floyd Patterson in 1952, recovered from a tentative start to win a 40-26 decision.
Khan looked nervous at first and trailed a much more experienced opponent after the first two rounds but he then moved into gear, shaking Yeleuov with flashing combinations.
The world junior champion, relying on swift footwork, rare hand speed and crowd-pleasing body moves, moved level on points in the third round.
With Yeleuov running out of steam, he then took total control of the bout. "I felt nervous in the dressing room before the contest," said Khan. "I knew I was behind but I thought he would tire down and that's what happened."
The only British boxer to have entered the tournament, Khan will face Cuba's Mario Kindelan in an intriguing final on Sunday.
Kindelan, the defending champion, who cruised past Russian Murat Khrachev in his semi-final, is widely regarded as the world's finest pound-for-pound amateur boxer and is nearly twice Khan's age at 33.
If he were to crown his campaign by toppling Kindelan, Khan would cause one of the greatest sensations in Olympic boxing history.
Khan is not the only boxer in Athens trying to earn a line behind Patterson in the record books.
Turkey's light-flyweight Atagun Yalcinkaya, who also qualified for the final on Friday, is six days younger than the Briton.