You may not get the taxes you paid on your winnings back from Pennsylvania, but your resident state of New Jersey will give you a credit for the tax you paid to Pennsylvania.
If you are a resident of New Jersey, and Pennsylvania state taxes were withheld on your jackpot winnings, you need to indicate that you 'earned income in another state' in the PERSONAL INFO section of TurboTax (see screenshot below).
Most likely you received a 1099-G to report your winnings, which you enter in your FEDERAL TAXES. The info will carry over into your state returns for New Jersey and Pennsylvania. You can type '1099-G' in the Search Window in the upper right corner of TurboTax to 'Jump to' the section for entering this.
Then in the STATE TAXES section, you will go thru the Pennsylvania interview for a non-resident, and verify the income and taxes paid to them. In the interview for a New Jersey resident, you will confirm the income and taxes paid to Pennsylvania.