We consider the surgical subtask of automated extraction of embedded suturing needles from silicone phantoms and propose a four-step algorithm consisting of calibration, needle segmentation, grasp planning, and path planning. We implement autonomous extraction of needles using the da Vinci Research Kit (dVRK). The proposed calibration method yields an average of 1.3mm transformation error between the dVRK end-effector and its overhead endoscopic stereo camera compared to 2.0mm transformation error using a standard rigid body transformation. In 143/160 images where a needle was detected, the needle segmentation algorithm planned appropriate grasp points with an accuracy of 97.20% and planned an appropriate pull trajectory to achieve extraction in 85.31% of images. For images segmented with >50% confidence, no errors in grasp or pull prediction occurred. In images segmented with 25-50% confidence, no erroneous grasps were planned, but a misdirected pull was planned in 6.45% of cases. In 100 physical trials, the dVRK successfully grasped needles in 75% of cases, and fully extracted needles in 70.7% of cases where a grasp was secured.