Origin
Today this phrase is popular primarily in the world of sports. Whenever a game is close and the time is running low, sports commentators are known for saying the game's coming down to the wire!
Where'd it come from though? It didn't come from football, or basketball, but horse-racing. The phrase originated sometime around the year 1889. Back then, if a race was close, and the horses were neck and neck crossing the finish line, it was difficult to tell who crossed first. They didn't have the convenient cameras with replay options that we use today. So to help them determine who finished first, they used a wire, which was hung up across the track above the finishing line. The wire did its job well, allowing people to see in a close race which horse crossed first.
An example of this saying being used in 1889 by Scribner's Magazine:
"As the end of the stand was reached Timarch worked up to Petrel, and the two raced down to the ‘wire,’ cheered on by the applause of the spectators. They ended the first half mile of the race head and head, passing lapped together under the wire, and beginning in earnest the mile which was yet to be traversed."
Examples of The Phrase Being Used
"That math test was hard, it was all greek to me."
"Listening to my mom yell at me was difficult, I couldn't make out what she was saying. It was all greek to me."
Note: For most old phrases, sayings, and idioms, finding the exact person or date in which a phrase was originated is impossible. What's provided is the earliest known date the phrase or saying was being used in.
Usually the phrase is a quote from an old newspaper, book, poem, play, etc. If it is already being used by a newspaper though, then the saying must already be common knowledge among the people of that time. This means the origin of the saying is older.
For instance, if someone or something from 1660 is being quoted using a particular phrase or saying, you can assume that the saying must be older, as it's already popular and being commonly used in 1660. How far back varies from phrase to phrase.
There are some cases though where determining the origin from an exact person is possible, however these are not common. In short, most of the origins for these sayings give you a date so you can get a rough idea on when these phrases were being used.