Westerlund 1 (Wd 1) is the most massive stellar cluster in the Galaxy and is associated with an extended region of TeV emission. Here, we report the results of a search for GeV γ-ray emission in this region. The analysis is based on ∼4.5 yr of Fermi-Large Area Telescope data and reveals significantly extended emission which we model as a Gaussian, resulting in a best-fitting sigma of σ S = (0 475 ± 0 05) and an offset from Wd 1 of ∼1°. A partial overlap of the GeV emission with the TeV signal as reported by the High Energy Stereoscopic System (HESS) is found. We investigate the spectral and morphological characteristics of the γ-ray emission and discuss its origin in the context of two distinct scenarios. Acceleration of electrons in a pulsar wind nebula provides a reasonably natural interpretation of the GeV emission, but leaves the TeV emission unexplained. A scenario in which protons are accelerated in or near Wd 1 in supernova explosion(s) and are diffusing away and interacting with molecular material seems consistent with the observed GeV and TeV emission, but requires a very high energy input in protons, ∼1051 erg, and rather slow diffusion. Observations of Wd 1 with a future γ-ray detector such as the Cherenkov Telescope Array (CTA) provide a very promising route to fully resolve the origin of the TeV and GeV emission in Wd 1 and provide a deeper understanding of the high-energy astrophysics of massive stellar clusters.