Prepare the pie crust according to the recipe, or unroll the store-bought crust into a 9-inch pie plate and flute edges.
Preheat oven to 375 degrees F.
Make the Cheesecake Layer: Using an electric mixer, beat the cream cheese and sugar on medium speed until light and fluffy, about 2 minutes. Add the eggs, lemon juice and vanilla extract, and beat on medium-low speed until smooth and creamy and no lumps remain. Pour the filling into the prepared crust.
Make the Cranberry Layer: In a small bowl, stir together the cranberry sauce, cornstarch and brown sugar; spoon over cream cheese mixture and gently spread into an even layer.
Make the Crumb Topping: Combine the flour and sugars in a medium bowl; drizzle with the melted butter and toss with a fork until evenly moistened and the mixture forms many large chunks with pea-sized pieces mixed throughout. Sprinkle the streusel evenly over the cranberry layer.
Bake until the crust and topping are golden brown, about 45 to 55 minutes. Cover loosely with foil during the last 10 minutes if needed to prevent over-browning. Cool on a wire rack for 1 hour, then refrigerate for at least 2 hours before serving. Leftovers can be stored in the refrigerator for up to 4 days.