The victory stretches Federer's impressive unbeaten run in Basel to 18 matches.
"It's been a magical week".
"This was nearly a turnaround week for me", the Swiss legend said.
"To come through and win again here in my hometown, never knowing if this might be your last time.it obviously means a lot to me and it becomes very emotional", he said later.
Federer started his slow recovery with a break for 4-5; he then saved set points to level at 6-all, winning a game in which he double-faulted twice in a row and saved a set point with an ace. The 37-year-old has reached the final the past 12 times he's played the event. Federer prior to taking to the ATP tour was a ballkid himself in Basel.
The Romanian aims for a first title after arriving at Basel without a career win over a top-10 opponent.
Federer, 37, spent nearly 90 minutes less on court Saturday than Copil, who has had to come through six matches to reach his second career ATP final. His second career runner-up finish, both this season, earned nearly 210,000 euros ($239,000).
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Copil entered the event at No. 93 on the ATP rankings.
"I'm starting to really get to understand the conditions and it was consistent, my most consistent match by far".
The first-set tiebreaker was likely to be crucial given Copil's career record of 8-52 when losing the opening set in top-tier matches.
It was Federer's fourth title of 2018, and if he plays the Paris indoors this coming week he will retain a mathematical chance to take the year-end No. 1 ranking at the ATP Finals.
It looked as though Simon's comeback was going to be short-lived as Federer sprinted to a 4-1 advantage in the deciding set. Federer celebrated with a shout and a pump of his right fist.
Federer broke in the second game of the final set but Simon refused to give in and replied in the seventh. He clinched the win when Krajinovic sent a backhand long.