From Longman Phrasal Verbs Dictionary (surprisingly poorly edited, BTW): In the 59th minute Van Den Hornet conjured up a magnificent goal that brought his team to victory. Had I not already read that phrase I would have put its beginning as On the 59th minute by analogy with on the 4th of July, for example. Which is correct? Various dictionaries I've tried to refer to regarding this issue proved to be of little help .
Using on rather than in is actually more the exception than the rule. We use in to mean "inside of" or "within the confines of" when referring to minutes, months, years and centuries:
- He scored in the twelfth minute of overtime.
- She plans to visit Paris in March.
- My son was born in 2005.
- The Mona Lisa was painted in the 16th century.
Although we do use on when referring to specific days:
- We will drive to Houston on Saturday.
- My daughter's birthday is on the 19th.
- Cinco de Mayo falls on a Friday this year.
- On the first day of Christmas, my true love gave to me...
"In the 59th minute" is a metaphor "[impossibly] before time expires" similar to "at the last moment" or "just in time" (also, "just in the nick of time").
Regrettably, some English speakers will use the construction, "on the 59th minute," despite this being wrong. It likely comes from the idea of "on the stroke of midnight," referring to when a chime sounds or the old cowboy/gun slinger phrase, "at high noon" which comingles the idea of location with an infintely precise moment you cannot live "within."
The idea of "in a minute" likely comes from the idea of the smallest increment of time that is practical to "pass through." while scientifically we measure time much more precisely than in single seconds, when it comes to living our lives, we generally do not. Thus, we can say, "at the last second" in the sense of a location or the smallest measureable moment in time. Philisophically speaking, things don't happen in our lives "inside" of a second. But since minutes are made up of seconds, things do happen "inside" of a minute. Thus, "in the 59th minute" is the correct way to express the idea.