This paper derives a novel initial alignment method for the strapdown inertial navigation system (SINS), which transforms the attitude alignment into an attitude estimation problem. The process model of the proposed initial alignment method by attitude estimation is established by decomposition of the attitude matrix. The measurement model is constructed based on a generalized velocity integration formula that can integrate the inertial measurements over certain fixed time intervals. The contributions of the work presented here are twofold. First, the attitude estimation-based structure enables the proposed method to estimate the gyroscope biases other than the attitude quaternion, which is celebrated for the low-cost SINS. The second is the application of the proposed generalized velocity integration formula to attenuate the accumulated errors in vector observations caused by the traditional velocity integration formula. Experimental road tests are performed with a low-cost SINS, which validate the efficacy of the proposed method.