  We present the physical properties of KIC 5621294 showing light and timing
variations from the ${\it Kepler}$ photometry. Its light curve displays partial
eclipses and O'Connell effect with Max II fainter than Max I, which was fitted
quite well by applying third-body and spot effects to the system. The results
indicate that the eclipsing pair is a classical Algol-type system with
parameters of $q$=0.22, $i$=76$^\circ$.8, and $\Delta$($T_{1}$--$T_{2}$)=4,235
K, in which the detached primary component fills about 77\% of its limiting
lobe. Striking discrepancies exist between the primary and secondary eclipse
times obtained with the method of Kwee \& van Woerden. These are mainly caused
by surface inhomogeneities due to spot activity detected in our light-curve
synthesis. The 1,253 light-curve timings from the Wilson-Devinney code were
used for a period study. It was found that the orbital period of KIC 5621294
has varied due to a periodic variation overlaid on a downward parabola. The
sinusoidal variation with a period of 961 d and a semi-amplitude of 22.5 s most
likely arise from a light-time effect due to a third component with a mass of
$M_3 \sin i_3$=46.9 M$\rm_{Jup}$, which is in good agreement with that
calculated from the light curve itself. If its orbital inclination is larger
than about 40$^\circ$, the mass of the circumbinary object would possibly match
a brown dwarf. The parabolic variation could not be fully explained by either a
mass transfer between the binary components or an angular momentum via magnetic
braking. It is possible that the parabola may only be the observed part of a
period modulation caused by the presence of another companion in a wider orbit.
