Lines' 56th minute effort curled into the top left-hand corner from 25 yards to give Vale a much deserved winner after Tom Pope has missed a first-half penalty.
Liam Ridehalgh, making his debut for the home side after arriving on emergency loan from Huddersfield, was given a straight red on the half hour for pulling back Chris Birchall in the area.
But Owain fon Williams, who had already made a couple of fine saves in the opening period, dived to his left to fingertip Pope's spot-kick around the post.
The home side, who are still without a win since August, were always going to be up against in the second period in the search for an equaliser.
Pope twice was denied by fon Williams but Vale were nearly made to pay for not putting the game to bed when Ash Taylor headed against the bar for the home side in injury time.