How short do you think you are going to be; how long were you short?

around $800 , I am an intern as an F-1 student started working this August. I corrected my Form on 13th November and got correct deduction in my payroll after that.

I actually made a mistake only in my State tax form. M-4 I guess. My federral tax was deducted properly . Only had no deductions in state tax

Well again, you probably won't owe a penalty, but if you want to make a payment, I would check your state tax department web site.  They probably also have electronic payments.  I would be concerned that something would go wrong if you try to get this fixed by filing the M-4 for a big withholding then filing another one to go back to normal.

Okay thanks. So is paying it while filing returns is the best option for this?

If you might owe a penalty under state law, you should pay early.  I don't know all the state laws, I only know that for the IRS, you would not owe a penalty.  If you over-pay the estimate, you will get the difference as a refund (be sure to indicate in your tax program that you made an estimated payment).  As long as you don't owe a penalty, I would keep my money in my own pocket as long as possible.

Also by the way, F-1 visa holders are considered non-residents.  You need to file a 1040-NR which Turbotax does not provide.  Turbotax partners with Sprintax, or you can find another provider on your own.
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I would probably just make a payment directly instead of trying to file a new W-4, get a huge amount withheld for 1 or 2 checks, then file another new W-4.  Too many opportunities for your employer to screw up.

You can make a payment at www.irs.gov/payments.  Select "estimated payment for 2017" as the reason.  Estimated payments for income earned from August-December are due January 15, so you aren't even considered late yet.

Or, skip the payment altogether and pay what you owe when you file your return.  As long as the amount is under $1000, you should not be charged a penalty for late payments.

I actually made a mistake only in my State tax form. M-4 I guess. My federral tax was deducted properly . Only had no deductions in state tax. So while filing my taxes later, I can pay the amount owed right?