Going to an international transit area of a US airport 24 hours before my departure to avoid 1 day overstay of a US visa?

The comment from @choster is right on it. Basically the airline passenger manifest submitted to immigration/CBP will show the departure date and time the airline left. So even if your airline allows 24 hour check in, it's not going to help you because your departure date will pop up on your electronic I-94 record showing you overstayed.

CASE 1: IF YOU ENTERED VIA VWP

Make sure you leave latest by the last day you're supposed to. Overstaying on VWP essentially means you will never be able to use VWP to enter again and will always have to go for a visa.

CASE 2: IF YOU ENTERED WITH A VISA

Now if your current visa is expiring and the cost to get a new ticket is too high, personally I might do the one day overstay. It shouldn't be too much of problem when you go for renewal if you have an otherwise strong immigration profile and history. Disclaimer: My older brother overstayed by a couple of days previously and had his renewal without a problem.

However if your visa is a long term multiple entry visa and it is not close to expiring, overstaying even by one day automatically voids it and you would have to apply for a new visa so I would think twice or thrice about it.

What the Visa Expiration Date Means

It is important to note that there are circumstances which can serve to void or cancel the period of visa validity. If you overstay the end date of your authorized stay, as provided by the CBP officer at a port-of-entry, or United States Citizenship and Immigration Services (USCIS), your visa will automatically void or cancel unless;

You have filed an application in a timely manner for an extension of stay or a change of status;

That application is pending and not frivolous;

You can file to extend your status by a week with a good reason however it will cost $370 and it takes up to three months or more to be adjudicated by which time you will likely have left, and immediately you leave the petition to to extend status becomes void.

CONCLUSION

As with most immigration matters, sticking religiously to the rules is the safest way to go so you don't have to rely too much on the leniency/discretion of immigration officers. Find the cheapest ticket you can get and exit before the last day of valid stay. If you know someone with a bunch of airline miles, you can get him/her to get you a one way award ticket for a very reasonable price, maybe one third or less of a return ticket.

As chister said, there is no international area, and no exit controls. The duration you stayed, if needed for a further visa, will be calculated by the flight you left on.

That means the only way to avoid the overstay is to leave early.

As a wild idea, maybe you can leave the US a day early, flying (or driving) to Canada or Mexico, buy a flight back to your airport, basically doing a 'transfer' onto your original flight. I don't know if this can even work - just as a wild idea, as said.