High-speed actuation of laser frequency1 is critical in applications using lasers and frequency combs2,3, and is a prerequisite for phase locking, frequency stabilization and stability transfer among optical carriers. For example, high-bandwidth feedback control of frequency combs is used in optical-frequency synthesis4, frequency division5 and optical clocks6. Soliton microcombs7,8 have emerged as chip-scale frequency comb sources, and have been used in system-level demonstrations9,10. Yet integrated microcombs using thermal heaters have limited actuation bandwidths11,12 of up to 10kilohertz. Consequently, megahertz-bandwidth actuation and locking of microcombs have only been achieved with off-chip bulk component modulators. Here we demonstrate high-speed soliton microcomb actuation using integrated piezoelectric components13. By monolithically integrating AlN actuators14 on ultralow-loss Si3N4 photonic circuits15, we demonstrate voltage-controlled soliton initiation, tuning and stabilization with megahertz bandwidth. The AlN actuators use 300nanowatts of power and feature bidirectional tuning, high linearity and low hysteresis. They exhibit a flat actuation response up to 1megahertz-substantially exceeding bulk piezo tuning bandwidth-that is extendable to higher frequencies by overcoming coupling to acoustic contour modes of the chip. Via synchronous tuning of the laser and the microresonator, we exploit this ability to frequency-shift the optical comb spectrum (that is, to change the comb's carrier-envelope offset frequency) and make excursions beyond the soliton existence range. This enables a massively parallel frequency-modulated engine16,17 for lidar (light detection and ranging), with increased frequency excursion, lower power and elimination of channel distortions resulting from the soliton Raman self-frequency shift. Moreover, by modulating at a rate matching the frequency of high-overtone bulk acoustic resonances18, resonant build-up of bulk acoustic energy allowsa 14-fold reduction of the required driving voltage, making it compatible with CMOS (complementary metal-oxide-semiconductor) electronics. Our approach endows soliton microcombs with integrated, ultralow-power and fast actuation, expanding the repertoire of technological applications of microcombs.