Blackman's strike maintained United's unbeaten run in League One this season.
James Baxendale's weak back-pass let in Dave Kitson, who was through on goal before being hauled down by Dean Holden. Referee Geoff Eltringham sent-off Holden and awarded a penalty, which was converted by Blackman.
United keeper George Long was off his line smartly to block Will Grigg as he looked to try and capitalise on a defensive lapse.
Long thwarted Grigg again towards the end of the half, making a superb save to keep out the striker's close-range shot following a low cross from Florent Cuvelier.
United pair Tony McMahon and Kevin McDonald both tried long-range efforts, the first going just over and the other being saved by Karl Darlow.
In a rare Walsall attack, Cuvelier shot early after finding some space, sending the ball wide. As the home side continued to hold the upper hand, Ryan Flynn saw his shot deflected wide and Shaun Miller had an effort saved.