Town suffered a blow before kick off when Cavell Coo limped back to the changing room and was replaced in the starting line-up by Luke Hinsley.
And they started the game slowly with Woodley duo Mike Clarke and Levi Kennedy firing the first couple of shots off target.
Town created a couple of chances, but Mark Peers scooped the ball over the bar from close range and David Brown was unable to reach a Michael Barnes cross which was volleyed clear by a defender.
Brown was denied by the keeper soon after while Himnsley's volley from the edge of the area after a Peers cross had been cleared flew well wide.
Both sides had opportunities as the break approached, but Sam Walker's free kick was turned round the post by Jon Kennedy and Ashley Stott was unable to pick out any of the three men in the box after latching onto a Damien Allen pass.
Russ Headley was denied by Lincoln Adams and Hinsley early in the second half before breaking the deadlock in the 54th minute when he got a fortunate ricochet off Kennedy and the ball bobbled over the line.
And the visitors thought they had doubled their lead two minutes later until Clarke's header was ruled out by the linesman's flag.
Kennedy had to dive full length to his left to keep out Mark Haslam's 35-yard piledriver just past the hour mark as Woodley continued to look the more likely to score.
Nigel Jemson had a couple of heading chances - the first went into the keeper's arms the second over the top - and Peers had a shot blocked for a corner.
Adams went even closer with 15 minutes remaining, his downward header from a Paul Evans corner forcing a fine save out of Connett.