Copil, who entered the tournament in Switzerland as a qualifier, broke world No. 5 Zverev in the final game of the match to secure a 6-3, 6-7 (6), 6-4 win, adding to his upset of Marin Cilic earlier in the week.
Just as Federer looked to be settling into form after almost 30 unforced errors in the opening set, more disruption ensued.
Federer will now aim to win number 100 by the end of the season. "Just walking out for a Basel finals was always my dream, already just playing on centre court was a thrill". It's been a major week for me in terms of records.
The Swiss now owns nine trophies from his hometown event and has won his last 20 matches at the tournament.
"I still can't believe I won, it's been an unbelievable week for me", Copil said.
"Maybe I had to win through fighting. maybe it was different than some of my past victories. It's always more fun when you can celebrate in a big group". That's what I need if I want to do well in London, to do my best there.
"Hopefully now I can push my career up with a new place in the rankings". It means the world to me.
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"The results show it as well, so I'm very happy".
Copil, 28, ranked 93rd in the world started the final as the overwhelming underdog.
"I've gained a a lot of confidence in myself".
"I'm starting to really get to understand the conditions and it was consistent, my most consistent match by far".
"I'd love to see now what kind of results I can have - but it went his way today, this is how it ended". The 37-year-old has reached the final the past 12 times he's played the event. The inspired Romanian's booming serves and lobs created tension St Jakobshalle but the local favourite won five straight games to complete another memorable win.
Roger was magnificent behind his first serve, winning 32 out of 34 points and saving three out of five break points, creating the same number of chances on the other side and converting three to wrap up the win in straight sets and made his crowd erupting in joy.