After tamely surrendering the first set in 20 minutes, Medvedev found himself trailing 5-1 in the second but managed to fend off a match point and hold serve.
The 32-year-old is the first South African to reach the ATP Finals in 23 years and follows in the footsteps of compatriot Wayne Ferreira who achieved the feat in 1995.
American Jimmy Connors (109) is the only player to have won more than 100 career titles in the professional era.
The Swiss needed 2-1/2 hours to finally break his opponent's resistance as he claimed his 18th successive win at the tournament, which he is aiming to win for the ninth time. "I'd like to say a few words to the people on the TV around the world", Federer said."Thank you so much for supporting me this week, it's been a magical week and I can't thank you enough".
Simon responded, however, breaking Federer three times in the second set to force a third. This week, he has beaten No. 6 Marin Cilic and Zverev, who is ranked fifth. "There were a lot of breaks, it was all up and down".
"I'm very happy. I think it was my best performance this week", Federer said.
Germany's Angela Merkel Says She Won't Seek Re-Election; Will Leave Party Role
Merkel has been in a hard position since September 2017, when a national election left her party without an overall majority. His CDU colleague Christian von Steffen was more blunt: 'We need a meaningful programme with a clear path and new faces'.
The Swiss crowd favourite was never able to comfortably take command against an opponent whom he had defeated in their last six meetings.
The win was a reversal of the first main draw match of Zverev's career when, as a 16-year-old wild card, he lost to Bautista Agut in Hamburg in 2013.
"But winning is fun, winning gives you confidence and that's what I'll need if I want to do well in London and beat the best over there, so I hope that this confidence I gained from here is going to help me there".
Federer got the key break early in the deciding set and reeled off five straight points when facing three break chances in the sixth game.
There was an early warning when Simon broke in the second game and raced to a 5-2 lead. But two unforced errors in a row handed Federer the match.