Graph Neural Networks (GNNs) have shown impressive performance in graph representation learning, but they face challenges in capturing long-range dependencies due to their limited expressive power. To address this, Graph Transformers (GTs) were introduced, utilizing self-attention mechanism to effectively model pairwise node relationships. Despite their advantages, GTs suffer from quadratic complexity w.r.t. the number of nodes in the graph, hindering their applicability to large graphs. In this work, we present Graph-Enhanced Contextual Operator (GECO), a scalable and effective alternative to GTs that leverages neighborhood propagation and global convolutions to effectively capture local and global dependencies in quasilinear time. Our study on synthetic datasets reveals that GECO reaches 169x speedup on a graph with 2M nodes w.r.t. optimized attention. Further evaluations on diverse range of benchmarks showcase that GECO scales to large graphs where traditional GTs often face memory and time limitations. Notably, GECO consistently achieves comparable or superior quality compared to baselines, improving the SOTA up to 4.5%, and offering a scalable and effective solution for large-scale graph learning.