Control of the microbunching instability is a fundamental requirement in modern high-brightness electron linacs, in order to prevent misleading responses of beam optical diagnostics and contamination in the generation of coherent radiation, such as free electron lasers. We present the first experimental demonstration of control and suppression of microbunching instability by means of particles' longitudinal phase mixing in a magnetic chicane. In the presence of phase mixing, the intensity of the beam-emitted optical transition radiation, which is used as an indicator of the instability gain at optical wavelengths, is reduced by one order of magnitude and brought to the same level provided, alternatively, by beam heating. The experimental results are in agreement with particle tracking and analytical evaluations of the instability gain. This article is extended to a discussion of applications of magnetic-phase mixing to the generation of quasicold high-brightness ultrarelativistic electron beams.