Hi, I doubt you got a blood infection this way. However, I would check to make sure your injury doesn't get infected. So, if you see swelling, redness and /0r pain then I would suspect an infection. As to getting in the blood if it bled then it probably got cleaned out except that ones mouth has lots of bacteria in it. So it is not a good idea to suck on the injury and rather just wash it. So keep an eye on the wound and if you get any symptoms, as I mentioned, or a high fever then I would see a doctor.
I don't think you are at risk any more then if you cut your finger outside gardening etc. As long as you washed the area well and are monitoring it for infection I am sure you will be fine. Millions of American's handle and prepare raw meat on a daily basis and I am sure many have also cut themselves while preparing it. So don't worry.