When your health insurance from your job begins, you would no longer qualify for subsidized health insurance from the exchange. At the point, you would simply discontinue your exchange health insurance and have your group insurance cover you.
You would notify your insurer that you wish to end your plan. They will end your coverage, and will no longer bill the government for whatever subsidies you were receiving. You will get a statement, Form 1095-A, about the subsidies that you received for your tax filing for that year. You will need that statement to complete Form 8962. A draft version of Form 8962 instructions is available at: http://www.irs.gov/pub/irs-dft/i8962--dft.pdf
Your employer will list your new health insurance on your W-2 form. So, you will have proof that you had coverage for the whole time period. (You are allowed a three-month gap, in any case, without having to pay a penalty.)
And congrats on the new job!