Discrepancy between resume and start date for job in background check - is this an issue?

I received an offer for a junior role at a bank that's basically my dream job. I'm going through the background check process and, through submitting the application, discovered a job I originally thought I had started in December actually started in January of the following year. This discrepancy by a couple weeks was an honest mistake, as I had been speaking with that firm for quite some time before I got an offer and started working there. It was a very small firm, too.

However, I had the prior year on my resume as I thought I had started in December. They obviously saw the discrepancy between the year on my resume and what I submitted.

I want to tell them it was an honest mistake but I'm also really, really worried that they're going to think I did it to lie or something. If it wasn't on my resume I wouldn't feel worried at all, but because I have the incorrect year on my resume I'm worried they'll think it was to deceive.

Am I blowing this out of proportion? Will they not care if I just say it was an honest mistake and that the information I submitted was accurate? I don't want this to be an issue or to have them think I'm trying to deceive them.