Modern web applications make extensive use of API calls to update the UI state in response to user events or server-side changes. For such applications, API-level testing can play an important role, in-between unit-level testing and UI-level (or end-to-end) testing. Existing API testing tools require API specifications (e.g., OpenAPI), which often may not be available or, when available, be inconsistent with the API implementation, thus limiting the applicability of automated API testing to web applications. In this paper, we present an approach that leverages UI testing to enable API-level testing for web applications. Our technique navigates the web application under test and automatically generates an API-level test suite, along with an OpenAPI specification that describes the application's server-side APIs (for REST-based web applications). A key element of our solution is a dynamic approach for inferring API endpoints with path parameters via UI navigation and directed API probing. We evaluated the technique for its accuracy in inferring API specifications and the effectiveness of the "carved" API tests. Our results on seven open-source web applications show that the technique achieves 98% precision and 56% recall in inferring endpoints. The carved API tests, when added to test suites generated by two automated REST API testing tools, increase statement coverage by 52% and 29% and branch coverage by 99% and 75%, on average. The main benefits of our technique are: (1) it enables API-level testing of web applications in cases where existing API testing tools are inapplicable and (2) it creates API-level test suites that cover server-side code efficiently while exercising APIs as they would be invoked from an application's web UI, and that can augment existing API test suites.