The incident early in the second half proved a major turning point, with Clough's team leading at the time courtesy of Chris Porter's 17th-minute goal.
Craig Westcarr scored from the spot after 52 minutes and recalled full-back James Chambers then notched his first ever goal in 328 league games to win it and lift the Saddlers to within a point of the League One play-off zone.
It was the Blades' first loss in nine matches, and Smith said: "They were not on an unbeaten run for nothing and are well organised. I kept being told that we don't come from behind to win, so I'm really pleased we have.
"To be fair, they started better than us for 20 minutes and if we had weathered the storm we would have been alright. But they scored a really good goal before we sorted their system out."