Laser-plasma accelerators outperform current radio frequency technology in acceleration strength by orders of magnitude. Yet, enabling them to deliver competitive beam quality for demanding applications, particularly in terms of energy spread and stability, remains a major challenge. In this Letter, we propose to combine bunch decompression and active plasma dechirping for drastically improving the energy profile and stability of beams from laser-plasma accelerators. Realistic start-to-end simulations demonstrate the potential of these postacceleration phase-space manipulations for simultaneously reducing an initial energy spread and energy jitter of ∼1-2% to ≲0.1%, closing the beam-quality gap to conventional acceleration schemes.