The Owls started brightly and Scott Doe was helpless in the 13th minute as he turned Gary Teale's well worked corner into his own net after challenging Giles Cole inside the visitors' six yard box and deflecting the ball past veteran goalkeeper Tony Roberts.
The Wednesday fans were soon celebrating a second when Clinton Morrison showed his predatory instincts two minutes later by smashing home first time from no more than six yards out after receiving possession from Chris Sedgwick's cushioned header.
Roberts would then deny Morrison a second soon after with a smart stop while Marcus Tudgay saw his glancing header tipped over before Wednesday took their foot off the gas to give Dagenham some hope of a recovery.
John Still's side enjoyed plenty of possession after the restart but rarely troubled Owls keeper Nicky Weaver until late on when substitute John Nurse steered a header wide from an excellent centre by Damien McCrory, before Peter Gain also went close as he saw a chipped effort come back off the bar.
Tudgay produced Wednesday's only moment of quality in the second-half with a header over with minutes remaining following a good cross by Sedgwick.