High-$z$ blazars (z $\geq 2.5$) are the most powerful class of persistent $\gamma$-ray sources in the Universe. These objects possess the highest jet powers and luminosities and have black hole masses often in excess of $10^9$ solar masses. In addition, high-$z$ blazars are important cosmological probes and serve as test objects for blazar evolution models. Due to their large distance, their high-energy emission typically peaks below the GeV range, which makes them difficult to study with Fermi/LAT. Therefore, only the very brightest objects are detectable and, to date, only a small number of high-z blazars have been detected with Fermi/LAT. In this work, we studied the monthly binned long-term $\gamma$-ray emission of a sample of 176 radio and optically detected blazars that have not been reported as known $\gamma$-ray sources in the 3FGL catalog. In order to account for false-positive detections, we calculated monthly Fermi/LAT light curves for a large sample of blank sky positions and derived the number of random fluctuations that we expect at various test statistic (TS) levels. For a given blazar, a detection of TS > 9 in at least one month is expected $\sim 15\%$ of the time. Although this rate is too high to secure detection of an individual source, half of our sample shows such single-month $\gamma$-ray activity, indicating a population of high-energy blazars at distances of up to z=5.2. Multiple TS > 9 monthly detections are unlikely to happen by chance, and we have detected several individual new sources in this way, including the most distant $\gamma$-ray blazar, BZQ J1430+4204 (z = 4.72). Finally, two new $\gamma$-ray blazars at redshifts of z = 3.63 and z = 3.11 are unambiguously detected via very significant (TS > 25) flares in individual monthly time bins.