Prevalent approaches in cross-lingual relation and event extraction use graph convolutional networks (GCNs) with universal dependency parses to learn language-agnostic representations such that models trained on one language can be applied to other languages. However, GCNs lack in modeling long-range dependencies or disconnected words in the dependency tree. To address this challenge, we propose to utilize the self-attention mechanism where we explicitly fuse structural information to learn the dependencies between words at different syntactic distances. We introduce GATE, a Graph Attention Transformer Encoder, and test its cross-lingual transferability on relation and event extraction tasks. We perform rigorous experiments on the widely used ACE05 dataset that includes three typologically different languages: English, Chinese, and Arabic. The evaluation results show that GATE outperforms three recently proposed methods by a large margin. Our detailed analysis reveals that due to the reliance on syntactic dependencies, GATE produces robust representations that facilitate transfer across languages.