In a large saucepan, combine apples, cranberries, cinnamon stick, sugar, and water; bring to a boil. Reduce heat; cover, and simmer, stirring occasionally, until apples are tender, 30 to 35 minutes. (If sauce begins to stick to the bottom of the pan, add 2 to 4 tablespoons more water.)
Remove from heat; discard cinnamon stick. Stir in lemon juice.