Place a strip of double-sided tape on all four back edges of the linoleum patch. Press it into place over the cut.
Use a utility knife to cut around the patch, using the edges as a template. Remove the patch and pull off the tape.
Push a putty knife under the damaged section that you cut around and pry it up. Scrape off any adhesive still stuck to the subfloor.
Apply a layer of linoleum adhesive to the back of the patch, according to the package directions. Press the piece into the spot where the damaged portion was removed so the patterns line up. Roll it flat with a hand roller.
Apply a bead of liquid seam filler around the four edges of the patch. Keep people and pets off the floor until the adhesive and seam filler are dry, according to the directions for the seam filler.