Azure Search is a search-as-a-service solution that allows developers and cloud architects to embed a sophisticated search experience into web and mobile applications without having to worry about the complexities of full-text search and without having to deploy, maintain, or manage any infrastructure.
With Azure Search, a developer can enable the power of searching data in their application, reduce the complexity around managing and tuning a search index, and get up and running quickly using familiar interfaces and a consistent platform.
Also,
Full Text Search is available in Preview for SQL Azure. See http://azure.microsoft.com/blog/2015/04/30/full-text-search-is-now-available-for-preview-in-azure-sql-database/