An 80th-minute goal - with outstanding Saddlers skipper Andy Butler beating County stopper Bartosz Bialkowski from Ben Purkiss' free-kick - was enough to give the visitors all three points.
Smith said: "They (County) are a very, very good team and played much better than us first half. Second half we stepped it up and tried to press further up the pitch and I thought we did a lot better, when we won the ball in their half we kept the ball better and looked dangerous.
"We were organised and made it hard for them to get in behind us and in between us and kept the ball in front of us. It will certainly give the lads confidence to come to a good side and come out victors."
There was some confusion over who got the winning goal after Butler's finish was made sure by substitute George Bowerman, although the former was credited with the strike.
Smith added: "There's an argument going on in the dressing room at the moment - it looks touch and go but it looks like it might be George Bowerman's, but I think Andy will have something to say about that as he said coming off the pitch that he always seems to score here."