Rinse the cranberries and pick out any soft or shriveled berries and stems.
Combine the sugar, orange juice, and water in a medium saucepan. Bring to a boil over medium-high heat, stirring until the sugar dissolves.
Add the cranberries, orange zest, cinnamon stick, and salt. Return to a boil, then lower the heat to a steady simmer.
Simmer, stirring occasionally, until most of the berries have popped and the sauce starts to thicken, about 10 to 12 minutes. Mash some of the berries with the back of a spoon for a jammier texture.
Remove from the heat and take out the cinnamon stick. The sauce will look loose but thickens a lot as it cools.
Cool for about 30 minutes, then transfer to an airtight container and refrigerate until chilled. Serve cold or at room temperature.
