First multi-wavelength photometric light curves (LCs) of the short-period eclipsing binary (EB) 1SWASP J034439.97+030425.5 (hereafter J0344) are presented and analyzed by using the 2013 version of the Wilson-Devinney (W-D) code. To explain the asymmetric LCs of J0344, a cool star-spot on the less massive component was employed. The photometric solutions suggest that J0344 is a W-subtype shallow contact EB with a contact degree of f = 4.9% +/- 3.0% and a mass ratio of q = 2.456 +/- 0.013. Moreover, an obvious third light was detected in our analysis. We calculated the average luminosity contribution of the third light to the total light, and that value reaches up to 49.78%. Based on the O - C method, the variations of the orbital period were studied for the first time. Our O - C diagram reveals a secular decrease superimposed on a cyclic oscillation. The orbital period decreases at a rate of dP/dt = -6.07 x 10(-7) d yr(-1), which can be explained by the mass transfer from the more massive component to the less massive one. Besides, its O - C diagram also shows a cyclic oscillation with an amplitude of 0.0030 d and a period about 7.08 yr, which can be explained by the presence of a third body with a minimum mass of M-3min = 0.15 +/- 0.02 M-circle dot. The third component may play an important role in the formation and evolution of J0344 by drawing angular momentum from the central system.