You have that backwards. Electrons move through an electric field so that they lose their potential: i.e. from a place of higher potential to a place of lower potential.
There are several useful examples to consider.
Suppose that the power source for the circuit is an object containing separated charges, such as a capacitor with it two plates that have opposite and equal charges. Electrons will flow from the (-) plate where there is a surplus of them through our circuit elements to the (+) plate where there is a dearth. As this happens, the voltage on the capacitor slowly goes down, until the plates no longer separate charges: the voltage is zero.
Why do we still call this a circuit when the electrons simply move from one place to another? Although no individual electron actually crosses the capacitor, it does look as if electricity is flowing across the capacitor simply because electrons are entering one side and at the same time leaving the other, and they are all identical: we cannot label our favorite electron and see whether it goes in one side and out the other. So the capacitor, and its load, do appear to form a complete circuit even though the capacitor is actually internally open.
Using a capacitor as a source of electricity is similar to water powered machine. We pour water into an upper reservoir, and it flows down from there, powering our machine via a turbine, and collects in some lower reservoir. In its descent, the water loses gravitational potential energy. When the upper reservoir is empty, the machine stops working. Someone has to come in and recharge the machine by doing work: transferring the water back to the upper reservoir, lifting it against gravity. (The gravitational analogy is not perfect, because there are no negative and positive masses, like there are negative and positive charges. The similarity is that it takes work to separate two masses to overcome gravity, and it takes work to separate opposite charges to overcome the attractive force caused by the field.)
Now, unlike a capacitor, a battery contains a chemical reaction which continuously produces a fresh separation of charges. (Incidentally, the word "battery" once referred to an array of capacitors, not to chemical cells!) With chemical cells, we no longer worry about running out of the small capacity of electrons stored in a plate, because a chemical reaction is replenishing them. Of course, the reaction will eventually reach equilibrium and stop. But that can take much longer. For example, alkaline batteries hold a lot more energy than capacitors. In a battery circuit, the same electron will go around multiple times: it is a true circular path. The spent electrons go back into the battery, where the chemical reaction carries them against the electric field back to the negative electrode, restoring their potential energy. The energy of the chemical reaction performs work on the electrons, transferring its energy to them.
We can also pump electricity continuously, using generators. By moving a coil in a magnetic field, we can keep inducing a voltage, forcing the electrons to keep going around and around in the circuit. This is like adding a pump to the water machine, so that someone can just turn a crank to pump the water back to the upper reservoir, allowing the water-powered machine to run continuously. A changing magnetic field forces the electrons to move inside the coil, so that a voltage develops and then the electrons flow through the rest of the circuit back into the other side of the coil. By fluctuating the magnetic field, we create a back and forth motion of electricity through the circuit (alternating current, AC). That by itself can be used as a source of power for many kinds of devices, and can be rectified to direct current (DC) for devices which require it.