With the rapid development of general cloud services, more and more individuals or collectives use cloud platforms to store data. Assured data deletion deserves investigation in cloud storage. In time-sensitive data storage scenarios, it is necessary for cloud platforms to automatically destroy data after the data owner-specified expiration time. Therefore, assured time-sensitive data deletion should be sought. In this paper, a fine-grained assured time-sensitive data deletion (ATDD) scheme in cloud storage is proposed by embedding the time trapdoor in Ciphertext-Policy Attribute-Based Encryption (CP-ABE). Time-sensitive data is self-destructed after the data owner-specified expiration time so that the authorized users cannot get access to the related data. In addition, a credential is returned to the data owner for data deletion verification. This proposed scheme provides solutions for fine-grained access control and verifiable data self-destruction. Detailed security and performance analysis demonstrate the security and the practicability of the proposed scheme.