We report the emergence of a narrow H₂O maser feature with an FWHM of 21 km s^-1 in the LINER NGC 1052, which has been known to show only a broad (FWHM > 100 km s^-1) maser line profile with relatively bright continuum radio emission. The new narrow maser feature with a peak flux density of 47 mJy at V_LSR = 1787 km s^-1 is redshifted by 328 km s^-1 with respect to the systemic velocity. Broad features with peak velocities of 1510 and 1704 km s^-1, more redward than ever observed before, are also detected. The profile of the new narrow feature possibly shows brightening by 16% ± 9% and narrowing by 30% ± 12% between 2003 May 30 and June 2. During the same time, the continuum flux density has increased by 21%. Synchronous variation of maser and continuum flux densities on a timescale of days resembles that in Mrk 348, which is also a broad megamaser source with a bright radio continuum. Continuum and maser brightening and narrowing indicate that an increase of the background seed photon and an increase of maser gain have occurred simultaneously. A jet component running behind a mixture of ionized regions and X-ray dissociation regions at a subrelativistic velocity can produce such short-time variation. Another explanation is an interaction between the jet and molecular clouds.