Since the state return cannot be efiled before the federal has been accepted if you waited until the last day to file it is possible the state did not get sent on the last day ... acceptance of the federal return can sometimes take days which can make the return technically late.

When you e-file your federal return you will receive two emails from TurboTax.  The first one will say that your return was submitted.  The second email will tell you if your federal return was accepted or rejected.  If you e-filed a state return, there will be a third email to tell you if the state accepted or rejected your state return.

Your return's electronic filing status history records all e-filing activity related to the return, including the Electronic Postmark. You can print it out if needed. 

TurboTax Online

https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp  -- enter the Social Security number of the first taxpayer on the return and the ZIP code to retrieve your filing status history.

TurboTax CD/Download software

(Windows) From the TurboTax File menu, select Electronic Filing > Show Electronic Filing Status History.
(Mac) From the TurboTax File menu, select Check E-Filing Status.

Thanks for the step-by-step instruction on e-filing status history, I have printed out a copy of the Electronic Postmark which clearly indicates I filed both Federal and State tax returns on time. As to the acceptance of the federal return, the e-filing status history also shows that mine was accepted before this year's filing deadline.

It is my understanding, however, tax return is considered to be filed as soon as e-file button is pressed and TurboTax Server acknowledges it has been received, irrelevant to the acceptance of the federal return. It should be exactly the same as printing out State tax return, and along with a copy of Federal tax forms mailing it out through postal service - it is possible that it takes days for Federal to accept the return, but as long as the postmark at the post office is on or before 7/15, the State cannot say it is filed late. Or even  in an extreme scenario there are errors in the Federal Tax return and it gets sent back, this would cause issues on the State tax return but it would not be a filing-late issue. Please correct me if I am wrong here. Would love to hear thoughts/opinion from other TurboTax users as well. Thanks!

This is a discussion you need to have with your state ... use the information in your hand to dispute their position.  Also remember there are 2 different penalties ... failure to file and/or failure to pay timely.  Read the notice carefully so you know what it is all about.