as the Wikipedia article you linked explains, capillary action is due to the "surface tension" (cohesion within the liquid) as well as "adhesion" (attraction between the water and the paper).
In particular, surface tension
is the energy per unit area of the surface of the liquid. The molecules of water inside the liquid are nicely sitting in a potential "hole" induced by the adjacent molecules.
However, the molecules of water that are on the surface lack the attractive force to the now non-existent neighbors on the side where the liquid no longer exists. Because attractive forces are related to a negative potential energy, the lack of them leads to a positive potential energy.
Consequently, water will contain some extra positive energy called the "surface tension". It's a form of potential energy but it can only be interpreted in this way at the molecular level. As hinted in the previous paragraph, the surface tension is a contribution to the energy that is proportional to the surface. That's why the surface tension causes water to take the shape of spherical droplets, among many other things, because the sphere minimizes the surface among shapes of the same volume.
Similarly, adhesion - between the paper and water in this case - adds a negative energy proportional to the surface of contact. Again, it may be explained by the (now added, not lacking) negative potential energy between the molecules of the water and the molecules of the paper that attract. To reduce the energy, water tries to touch the paper as much as it can.
Only the total energy is conserved. The energy needed to lift the water is obtained from the surface tension and adhesion. Microscopically, you may literally imagine that the molecules of water at the top were attracted by the nearby (and so far higher) molecules of paper.