Finite-time stable guidance and feedback control scheme for steering a class of autonomous underactuated vehicles in SE(3), is given here. The underactuated vehicles are characterized by fewer control inputs than the number of configuration variables and modeled as a rigid body with four control inputs. These control inputs actuate the three degrees of freedom of rotational motion and one degree of freedom of translational motion in a vehicle body-fixed coordinate frame. This actuation model is appropriate for a wide range of underactuated vehicles including spacecraft with internal attitude actuators, vertical take-off, and landing (VTOL) aircraft, fixed-wing multirotor unmanned aerial vehicles (UAVs), maneuverable robotic vehicles, etc. The guidance and control problems are developed on the special Euclidean group of rigid body motions, SE(3), in the framework of geometric mechanics, which represents the vehicle dynamics globally on this configuration manifold. The desired position and attitude in SE(3) is tracked by the finite-time stable translational and attitude controller developed here. The overall stability analysis of the feedback system is addressed. Discrete time models for the dynamics and control schemes of the UAV are obtained in the form of Lie group variational integrators using the discrete Lagrange-d'Alembert principle. Almost global finite-time stability of the overall feedback system over the state space is demonstrated and the importance of finite-time stability is presented through numerical simulations.