Many order fulfillment applications in logistics, such as packing, involve picking objects from unstructured piles before tightly arranging them in bins or shipping containers. Desirable robotic solutions in this space need to be low-cost, robust, easily deployable and simple to control. The current work proposes a complete pipeline for solving packing tasks for cuboid objects, given access only to RGB-D data and a single robot arm with a vacuum-based end-effector, which is also used as a pushing or dragging finger. The pipeline integrates perception for detecting the objects and planning so as to properly pick and place objects. The key challenges correspond to sensing noise and failures in execution, which appear at multiple steps of the process. To achieve robustness, three uncertainty-reducing manipulation primitives are proposed, which take advantage of the end-effector's and the workspace's compliance, to successfully and tightly pack multiple cuboid objects. The overall solution is demonstrated to be robust to execution and perception errors. The impact of each manipulation primitive is evaluated in extensive real-world experiments by considering different versions of the pipeline. Furthermore, an open-source simulation framework is provided for modeling such packing operations. Ablation studies are performed within this simulation environment to evaluate features of the proposed primitives.