Flutter's Pub package manager allows developers to easily integrate third-party dependencies across multiple platforms, making it a production-ready tool for building cross-platform apps. The process of adding dependencies involves specifying the dependency in the pubspec.yaml file and then running the 'flutter pub add' command to download the latest version of the package. This can be done using either the pub.dev repository or by downloading from local paths or Git repositories. Sentry, a mobile performance monitoring tool, was successfully integrated into a Flutter project using this process. Once added, dependencies can be easily imported and used in code, with automatic updates available through the 'flutter pub upgrade' command. Overall, integrating third-party dependencies in Flutter is a seamless process that confirms its status as a versatile development tool.