With the advantages of small size, low cost, and moderate accuracy, an open-loop fiber-optic gyroscope (FOG) has a wide range of applications around control and automation. For the most cost-sensitive applications, a simple and stable digital algorithm with a reduced control-circuit volume and cost is highly desirable to realize high-precision control of a FOG. In this work, a new algorithm for an open-loop FOG is proposed based on the discrete multi-point demodulation in the sinusoidal modulation period. Utilizing this algorithm, stable control and angular velocity calculation of a gyro are realized with effectively suppressed gyro error. The use of this algorithm greatly reduces the requirements for processing power and simplifies the gyro circuit. Based on this algorithm, a digital FOG with a volume of only 25 × 20 × 40 mm3 achieves a bias instability of less than 0.15°/h, an angle random walk (ARW) of less than 0.015°/√h, a start-up time of less than 1 s, and a 3 dB bandwidth beyond 160 Hz. This low-cost, compact, and high-performance gyro is sufficient to satisfy the requirements of applications in the navigation and control fields such as unmanned driving.