Periodograms are common tools used to search for periodic signals in unevenly spaced time series. The significance of periodogram peaks is often assessed using false alarm probability (FAP), which in most studies assumes uncorrelated noise and is computed using numerical methods such as bootstrapping or Monte Carlo. These methods have a high computational cost, especially for low FAP levels, which are of most interest. We present an analytical estimate of the FAP of the periodogram in the presence of correlated noise, which is fundamental to analyze astronomical time series correctly. The analytical estimate that we derive provides a very good approximation of the FAP at a much lower cost than numerical methods. We validate our analytical approach by comparing it with Monte Carlo simulations. Finally, we discuss the sensitivity of the method to different assumptions in the modeling of the noise.