Wakefield showed more attacking intent in the early stages than many sides to have visited the Shay this season, without ever threatening to score.
At the other end, efforts from distance by Tom Baker and Cavell Coo both forced decent saves from for
mer Halifax stopper Woods.
Damien Allen fired wide from a Nigel Jemson nod down and Junior Brown had a dipping volley tipped over by Woods as they looked for the opener.
Wakefield were reduced to 10 men after 28 minutes when Woods handled outside his area and was red carded.
Striker James Walshaw took the keeper's shirt and was almost beaten by the resulting Baker free kick which thudded back off a post.
But other than that, Town failed to trouble the stand-in keeper until two minutes after the break when Jemson fired home a right wing cross from 10 yards.
Baker doubled Town's advantage in the 58th minute, firing home a dipping volley from 30 yards, but within 60 seconds Chris Howarth pulled one back from the edge of the area.
Wakefield put defender Caldon Henson in goal and his first task was to pick the ball out of the net as Jemson bagged his second with a header from a 67th-minute Allen corner.
Baker was denied a second goal by a full length save from Henson with 14 minutes remaining but Wes Milnes ensured a tense finale when he poked home from an 85th minute corner.
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