You could try posting this to /r/etymology. This phrase is compositional in meaning, i.e. its meaning is the sum of its parts. It seems to have been used at least since the late 18th century. The earliest cite I could find is from an anonymous letter published in 1767:

Willing to save the day by his own troops perhaps, he delayed too long, or too much hesitated at the precision of his orders.

Likely to have been inspired by the more common phrases (at the time) "won the day" or "lost the day."