<p>The paper presents a computationally efficient convex optimization-based strategy for tackling the eco-driving problem. This strategy facilitates the co-optimization of velocity planning and powertrain control, offering an approach to enhance fuel efficiency. Discrete gear optimization results in the traditional eco-driving problem often being formulated as a hard-to-solve mixed-integer (non-convex) nonlinear optimal control problem (OCP). Through several reformulations and relaxations, the original non-convex eco-driving optimization problem is transformed into a convex optimal control problem with global optimality guarantees. A rigorous proof of the equivalence between the original non-convex and the convexified problem is provided. The proposed approach enables optimization of the engine torques, gear choices, and speed profiles that can be efficiently solved in a single optimization problem. Real-time iteration sequential quadratic programming (SQP) serves to rapidly compute accurate solutions of our formulated convex OCP. Simulation and hardware-in-the-loop experiments on the embedded vehicle controller demonstrate both a superior fuel-saving performance compared to existing methods and real-time feasibility.</p>