We report the observation of the baryonic decay $\bar{B}^0 \to \Lambda^+_c \bar{p} K^- K^+$ using a data sample of $471 \times 10^6$ $B\bar{B}$ pairs produced in $e^+e^-$ annihilations at $\sqrt{s}=10.58 \mathrm{\,GeV}$. This data sample was recorded with the $BABAR$ detector at the PEP-II storage ring at SLAC. We find ${\cal B}\left(\bar{B}^0 \to \Lambda^+_c \bar{p} K^- K^+ \right) = \left(2.5 \pm 0.4_{(\text{stat})} \pm 0.2_{(\text{syst})}\pm 0.6_{{\cal B}\left(\Lambda^+_c\right)} \right) \times 10^{-5}$, where the uncertainties are statistical, systematic, and due to the uncertainty of the $\Lambda^+_c \to p K^- \pi^+$ branching fraction, respectively. The result has a significance corresponding to 5.0 standard deviations, including all uncertainties. For the resonant decay $\bar{B}^0 \to \Lambda^+_c \bar{p} \phi$, we determine the upper limit ${\cal B}\left(\bar{B}^0 \to \Lambda^+_c \bar{p} \phi \right) <1.2 \times 10^{-5}$ at $90\%$ confidence level.