We present an analysis of the transit timing variations (TTVs) in the multi-transiting planetary system around Kepler-51 (KOI-620). This system consists of two confirmed transiting planets, Kepler-51b ($P_{\rm b} = 45.2\,\mathrm{days}$) and Kepler-51c ($P_{\rm c} = 85.3\,\mathrm{days}$), and one transiting planet candidate KOI-620.02 ($P_{02} = 130.2\,\mathrm{days}$), which lie close to a $1:2:3$ resonance chain. Our analysis shows that their TTVs are consistently explained by the three-planet model, and constrains their masses as $M_{\rm b} = 2.1_{-0.8}^{+1.5} M_{⊕}$ (Kepler-51b), $M_{\rm c} = 4.0 ± 0.4 M_{⊕}$ (Kepler-51c), and $M_{02} = 7.6 ± 1.1 M_{⊕}$ (KOI-620.02), thus confirming KOI-620.02 as a planet in this system. The masses inferred from the TTVs are rather small compared with the planetary radii based on the stellar density and planet-to-star radius ratios determined from the transit light curves. Combining these estimates, we find that all the three planets in this system have among the lowest densities yet determined, $ρ_p \lesssim 0.05\,{\rm g\,cm^{-3}}$. With this feature, the Kepler-51 system serves as another example of low-density compact multi-transiting planetary systems. We also identify a curious feature in the archived Kepler light curve during the double transit of Kepler-51b and KOI-620.02, which could be explained by their overlapping on the stellar disk (a planet-planet eclipse). If this is really the case, the sky-plane inclination of KOI-620.02's orbit relative to that of Kepler-51b is given by $Δ Ω = -25.3_{-6.8}^{+6.2}\mathrm{deg}$, implying significant misalignment of their orbital planes. This interpretation, however, seems unlikely because such a event that is consistent with all of the observations is found to be exceedingly rare. ; Comment: 9 pages, 4 figures, accepted for publication in ApJ