We establish a tight connection between two models of the λ-calculus, namely Milner's encoding into the π-calculus (precisely, the Internal π-calculus), and operational game semantics (OGS). We first investigate the operational correspondence between the behaviours of the encoding provided by π and OGS. We do so for various LTSs: the standard LTS for π and a new 'concurrent' LTS for OGS; an 'output-prioritised' LTS for π and the standard alternating LTS for OGS. We then show that the equivalences induced on λ-terms by all these LTSs (for π and OGS) coincide. These connections allow us to transfer results and techniques between π and OGS. In particular we import up-to techniques from π onto OGS and we derive congruence and compositionality results for OGS from those of π. The study is illustrated for call-by-value; similar results hold for call-by-name.