As long as your own web app is the only program which accesses that API, API changes don't need to be backwards compatible, since you will update your single page app immediately.
But as soon as you publish an API to the world, you need to care strictly for backwards compatibility, otherwise the other devs writing programs which consume your API will become very unhappy if their programs stop working with every new release of your API.
So either you go the route described by @RobertHarvey in his excellent answer, or you make some stable parts of the API public, and keep other parts internal.