26 pages, 10 captioned figures, 3 tables, authors from page 17, submitted to PRC, figures at http://aliceinfo.cern.ch/ArtSubmission/node/3363 ; see paper for full list of authors ; The production of K$^{*}(892)^{0}$ and $ϕ(1020)$ mesons in proton-proton (pp) and lead-lead (Pb-Pb) collisions at $\sqrt{s_\mathrm{NN}} =$ 2.76 TeV has been analyzed using a high luminosity data sample accumulated in 2011 with ALICE detector at the Large Hadron Collider (LHC). Transverse momentum ($p_{\mathrm{T}}$) spectra have been measured for K$^{*}(892)^{0}$ and $ϕ(1020)$ mesons via their hadronic decay channels for $p_{\mathrm{T}}$ up to 20 GeV/$c$. The measurements in pp collisions have been compared to model calculations and used to determine the nuclear modification factor and particle ratios. The K$^{*}(892)^{0}$/K ratio exhibits significant reduction from pp to central Pb-Pb collisions, consistent with the suppression of the K$^{*}(892)^{0}$ yield at low $p_{\mathrm{T}}$ due to rescattering of its decay products in the hadronic phase. In central Pb-Pb collisions the $p_{\mathrm{T}}$ dependent $ϕ(1020)/π$ and K$^{*}(892)^{0}$/$π$ ratios show an enhancement over pp collisions for $p_{\mathrm{T}}$ $∼$3 GeV/$c$, consistent with previous observations of strong radial flow. At high $p_{\mathrm{T}}$, particle ratios in Pb-Pb collisions are similar to those measured in pp collisions. In central Pb-Pb collisions, the production of K$^{*}(892)^{0}$ and $ϕ(1020)$ mesons is suppressed for $p_{\mathrm{T}}> 8$ GeV/$c$. This suppression is similar to that of charged pions, kaons and protons, indicating that the suppression does not depend on particle mass or flavor in the light quark sector.