If "I" were to do this...
I would sterilize my jars by placing them upright in a pot, fill the pot with hot water until about an inch or so above the top of the jars. Bring to a boil and boil for 10 minutes or more (this will depend on your altitude. Google is your friend here). Along with the jars, drop a pair of tongs, tong down, into the boiling water so they are sterilized also. Turn off the heat & carefully remove the jars with the tongs...it isn't easy emptying the hot water out of them with tongs so go very SLOW!!! Place empty jars upright on a sheet pan in a 200 degree oven (jars should be hot when the liquid is added). Once the jars are out of the pot, drop in the lids & rings. You do not want to boil these so be sure the water has cooled to below the boil. Let them sit in the water for 5-10 minutes.
While all this is happening you should have a pot on the range with your sauce. You probably don't want to boil it, but get it hot...very hot. If it bubbles, immediately reduce the heat, put a lid on it, and keep it HOT.
When the sauce is ready (hopefully it was ready as you were getting the jars sterilized), retrieve the lids & rings from the water pot, and gingerly remove the jars from the oven. Pour sauce into jars, place lids on top, and screw on rings. Let cool and you'll hear the lids pop as they vacuum themselves down.
You could pressure can, but really you don't have to. If everything is hot, they'll be fine...it's barbecue sauce which is typically a safe food anyway.