Put berries, sugar, lemon zest and juice, and 3/4 cup water into a wide pot. Cook over low heat, stirring occasionally, until fruit releases juices, about 30 minutes.
Smash berries with a potato masher. Cook until juices have thickened, stirring occasionally, about 45 minutes more.
Strain syrup into a 2-qt. glass measuring cup. Press fruit with a ladle to push remaining juice into cup. Discard seeds and pulp.
Pour strained syrup into jars, leaving 1/2 in. of headspace. Following Sunset canning instructions, process 10 minutes.
