One thing with a space elevator that makes it so much more efficient than rockets is precisely because you don't necessarily need the payload itself to supply this horizontal acceleration. The space elevator is attached to the ground at one end, and the other is way up in orbit. There must be forces in play _already_ for the entire thing to stay standing, before you get to any concept of a payload/car. Part of the idea of building the elevator in the first place is to solve for these orbital forces in a generalized way independent of the payloads themselves. It's like strapping various sized rockets to your various specific payloads, versus building a generalized model of a rocket ship, and then just putting the various payloads inside the generalized rocket ship. Space elevator is a further evolution of the concept. You don't even need to use the rocket ship abstraction anymore. You're generalizing/abstracting the orbital transition itself into the structure of the elevator, and then just send things up and down it. The payload now only needs to worry about moving along the elevator, the elevator itself has already "solved" for the orbital horizontal acceleration by nature of its structure existing in the first place.

In terms specifically of mass/energy conservation, as the other reply said, energy is borrowed from either the earth's rotation and/or kinetic energy from a counterweight at the end of the elevator up in orbit.