The IRS rules for the charitable contribution deduction are in IRS Publication 526.
Pub 526 has a section called "When To Deduct" that explains the details of placing your donation in a specific tax year. First, the general guideline:
Time of making contribution. Usually, you make a contribution at the time of its unconditional delivery.
After this statement, it gives some details for specific types of payments. Some of them that might apply to your last-minute donation are:
Checks. A check you mail to a charity is considered delivered on the date you mail it.
Text message. Contributions made by text message are deductible in the year you send the text message if the contribution is charged to your telephone or wireless account.
Credit card. Contributions charged on your bank credit card are deductible in the year you make the charge.
Pay-by-phone account. Contributions made through a pay-by-phone account are considered delivered on the date the financial institution pays the amount. This date should be shown on the statement the financial institution sends you.
For checks and text messages, it seems to me that the time you would use would be the local time when/where you send it.
For a credit card donation, local time would also probably apply, except that your credit card donation receipt is your proof that you donated, and you probably should use the time/date on the receipt.
For pay-by-phone account, the guidelines specifically state that you need to use the date on your statement.
Remember that your cash donation must have a record to substantiate it in order to be deductible: either a bank record, a receipt from the organization, or a paystub (if donated by payroll deduction). You are safest if the tax year you choose matches your record.