Graphene nanoplatelets (GNPs) were synthetized from graphite powder and, thereafter, embedded in poly(ethylene-co-vinyl alcohol) (EVOH) fibers by electrospinning in the 0.1-2 wt.-% range. The morphological, chemical, and thermal characterization performed on the electrospun nanocomposite fibers mats revealed that the GNPs were efficiently dispersed and rolled along the EVOH fibrilar matrix up to contents of 0.5 wt.-%. Additionally, the dielectric behavior of the nanocomposite fibers was evaluated as a function of the frequency range and GNPs content. The obtained results indicated that their dielectric constant rapidly decreased with the frequency increase and only increased at low GNPs loadings while the nanocomposite fiber mats became electrically conductive, with the maximum at 0.5 wt.-% GNPs content. Finally, the electrospun mats were subjected to a thermal post-treatment and dark films with a high contact transparency were obtained, suggesting that the nanocomposites can be used either in a nonwoven fibers form or in a continuous film form. This study demonstrates the potential of electrospinning as a promising technology to produce GNPs-containing materials with high electrical conductivity that can be of potential interest in intelligent packaging applications as 'smart' labels or tags.