The first-half was a hard fought one with both teams having chances.
The best went to Glenrothes when Lyall Shaw ran into the box past several defenders, but the shot was deflected wide and out for a corner.
At the other end, the solid Glens defence saw off any Tayport pressure.
Glens almost went ahead on 58 minutes after a Sean Simpson free-kick was pushed to the other side of the post by the Tayport goalkeeper. Tayport almost took the lead late on after the referee gave them a free-kick just outside the box.
The shot came to nothing and it was deflected out for a corner kick.
From the corner, a Tayport player sent in a shot and it just went wide of the goalpost.
Neither side could find the opening.
Glenrothes: Roberts, Napier, Lee, Mcbride, Shields, McKie, Gay, Simpson, Shaw, Martin, Trialist. Substitutes -
Trialist, Crombie, Moffat.