The ALICE experiment has measured low-mass dimuon production in pp collisions at \sqrt{s} = 7 TeV in the dimuon rapidity region 2.5y\eta^{}, \phi) into muons and semi-leptonic decays of charmed mesons. The measured production cross sections for \omega and \phi are \sigma_\omega (1p_t5 GeV/c,2.5y4) = 5.28 +- 0.54 (stat) +- 0.50 (syst) mb and \sigma_\phi(1p_t5 GeV/c,2.5y4)=0.940 +- 0.084 (stat) +- 0.078 (syst) mb. The differential cross sections d^2\sigma/dy dp_t are extracted as a function of p_t for \omega and \phi. The ratio between the \rho and \omega cross section is obtained. Results for the \phi are compared with other measurements at the same energy and with predictions by models.