The England Under 21 keeper made some early saves before Owen's 21st minute goal - a rebounded shot off Bob Peeters in the Arnhem goal.
It was Owen's 19th goal in European competition and one less than Ian Rush's record.
As the second half progressed Liverpool took their foot off the gas and the visitors were unlucky not to have equalised, but Kirkland made a few excellent blocks to deny the Dutch team.
Liverpool are now through to the fourth round of the competition to be drawn on Friday afternoon.
Liverpool: Kirkland, Carragher, Henchoz, Hyypia, Traore, Gerrard, Smicer, Diao, Murphy, Owen, Diouf. Subs: Dudek, Baros, Babbel, Hamann, Riise, Biscan, Mellor.
Vitesse: Jevric, Cornelisse, Stefanovic, Jansen, Frankel, Levtsjenko, Amoah, Janssen, Claessens, Peeters, Mbamba. Subs: Van Fessem, Mustapha Riga, Van Beukering, Dingsdag, Sone, Rojer.
Referee: Wolfgang Stark (Ergolding)