It was a cagey first half at Solitude, with clear-cut chances at a premium for both teams.
The best opportunity arrived on 20 minutes as Rory Hale won possession back for the Reds high up the pitch and fed Ryan Curran, who squared for McWoods, but his strike was beaten away by Chris Johns.
At the other end, Kirk Millar lifted an effort over the crossbar and Charlie Allen headed wide in the box in some of Linfield's half-chances.
It was an equally tentative second half with Odhran Casey heading over the crossbar from Hale's corner soon after the restart.
Then, just after the hour mark, Fitzpatrick spurned a glorious opportunity as he fired a low shot wide of the far post after the ball fell kindly to him in the box.
Soon after, Sam Roscoe glanced wide after meeting substitute Callumn Morrison's cross.
Substitute Gormley thought he had won the game in the last minute of injury time as his deflected strike rattled the crossbar.