Ultra-low noise lasers are essential tools in various applications, including data communication, light detection and ranging (LiDAR), quantum computing and sensing, and optical metrology. Recent advances in integrated photonics, specifically the development of the ultra-low loss silicon nitride ($Si3N4$) platform, have allowed attaining performance that exceeds conventional legacy laser systems, including the phase noise of fiber lasers. This platform can be combined with the monolithic integration of piezoelectric materials, enabling frequency-agile low-noise lasers. However, this approach has not yet surpassed the trade-off between ultra-low frequency noise and frequency agility. Here, we overcome this challenge and demonstrate a fully integrated laser based on the $Si3N4$ platform with frequency noise lower than that of a fiber laser while maintaining the capability for high-speed modulation of the laser frequency. The laser achieves an output power of 30 mW with an integrated linewidth of 4.3 kHz and an intrinsic linewidth of 3 Hz, demonstrating phase noise performance that is on par with or lower than commercial fiber lasers. Frequency agility is accomplished via a monolithically integrated piezoelectric aluminum nitride micro-electro-mechanical system (MEMS) actuator, which enables a flat frequency actuation bandwidth extending up to 400 kHz. Such a MEMS device is one of the largest fabricated structures, featuring MHz-level bandwidth, which is significantly higher than the typical kHz-level bandwidth of similarly sized mm-scale MEMS devices. The chirp nonlinearity of the frequency-modulated output reaches 0.08% without any linearization or pre-distortion, making it compliant with the requirement for long-range FMCW LiDAR. This ultra-low noise and frequency-agility combination is a useful feature enabling tight laser locking for frequency metrology, fiber sensing, and coherent sensing applications. Our results demonstrate the ability of “next generation” integrated photonic circuits (beyond silicon) to exceed the performance of legacy laser systems in terms of coherence and frequency actuation.