Congratulations! You have just crossed the first line towards success. It requires tremendous courage to come out and openly say, "I made a mistake," but more importantly, to say, "I want to change for the better." This one move and the lessons that I hope you have learned in this phase will be the foundation for your success in the future.
Go ahead, accept the offer. You are a graduate now, and hence you would not be lying. If an offer stage is already reached, then I presume that they have already interviewed you, so the possibility of them inquiring about why an organization recruited a non-graduate is very remote.
Just be sure: Never ever repeat what you have done! You are lucky that you have got an opportunity to move forward... It could have been worse...
Be thankful for this blessing and move forward.
Cheers,
Navneet