The Gaia-ESO Survey (GES) is a large spectroscopic survey that provides a unique opportunity to study the distribution of spectroscopic multiple systems among different populations of the Galaxy. We aim at detecting binarity/multiplicity for stars targeted by the GES from the analysis of the cross-correlation functions (CCFs) of the GES spectra with spectral templates. We develop a method based on the computation of the CCF successive derivatives to detect multiple peaks and determine their radial velocities, even when the peaks are strongly blended. The parameters of the detection of extrema (DOE) code have been optimized for each GES GIRAFFE and UVES setup to maximize detection. This code therefore allows to automatically detect multiple line spectroscopic binaries (SBn, n>1). We apply this method on the fourth GES internal data release and detect 354 SBn candidates (342 SB2, 11 SB3 and even one SB4), including only 9 SB2 known in the literature. This implies that about 98% of these SBn candidates are new (because of their faint visual magnitude that can reach V=19). Visual inspection of the SBn candidate spectra reveals that the most probable candidates have indeed a composite spectrum. Among SB2 candidates, an orbital solution could be computed for two previously unknown binaries: 06404608+0949173 (known as V642 Mon) in NGC 2264 and 19013257-0027338 in Berkeley 81. A detailed analysis of the unique SB4 (four peaks in the CCF) reveals that HD 74438 in the open cluster IC 2391 is a physically bound stellar quadruple system. The SB candidates belonging to stellar clusters are reviewed in detail to discard false detections. We warn against the use of atmospheric parameters for these system components rather than by SB-specific pipelines. Our implementation of an automatic detection of spectroscopic binaries within the GES has allowed an efficient discovery of many new multiple systems.