Yes, you will be entitled to a credit. When you file your state returns, you will want to start by filing the New York return first. You will be filing a part-year return for New York, but you will pay attention to the questions that ask about how much income you earned as a nonresident. That would be the amount you earned while still living in NJ. NY taxes all of your income: part as a resident, and part as a nonresident.
Then you prepare the NJ part-year resident return. Early in that return, you will be asked if you need to exempt income for the NJ return. Exempt the NY income. Because of reporting and programming requirements, your employer reports your separate state lines with this information, and TurboTax will add both the NY and NJ lines to calculate NJ tax, leading to a much higher tax due. But by checking the box in the screenshot below, you will remove the reduntantly reported income. Later, you are asked about double-taxed income. If you have prepared the NY return correctly, TurboTax will have calculated the credit for taxes paid to NY on your NJ return, reducing and possibly eliminating any tax owed.
See the screenshot below that shows the key screen entry on the NJ return:
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