To prepare for processing, bring the 2-piece canning lids to a simmer in a small saucepan. Turn heat off and keep warm. Fill your water bath canner half of water and bring to a boil. Bring a teakettle of water to a boil to add to your canner, if necessary.
Using a funnel and ladle, fill warm canning jars with hot applesauce one at a time, leaving 1/2 inch headspace. Remove any trapped air bubbles with a rubber spatula. Wipe the jar rim with a clean, damp towel or paper towel after filling. Place a warm flat jar lid on the jar, followed by a screw band. Turn the screw band to tighten the lid, but do not overtighten. Place each jar in the elevated rack of your water bath canner as it is filled.
Once all jars are filled and in the canner, lower the elevated rack. If the water does not cover the jars by 1 -2 inches, top off the canner with additional boiling water. Place the lid on the canner and bring to a full boil. Begin timing the processing time of 20 minutes for pints and quarts.
After 20 minutes, turn the heat off, remove the lid, and let the canner rest for 10 minutes. Lift the elevated rack and move jars to a towel-lined countertop to cool. Once jars are completely cool, check the lids for a good seal. Remove the screw bands and check that the center of each lid is pressed down. If you can press the center of the lid down the jar is not sealed and should be refrigerated and eaten within a few days.
For additional information on water bath canning, see this Introduction to Home Canning or visit the National Center for Home Food Preservation.