The defender miscued an attempt at a headed clearance following a punt by Vale goalkeeper Sam Johnson four minutes from time and inadvertently sent the ball looping over Boaz Myhill in the West Brom net.
Watched on by new defensive signing Jason Davidson, West Brom made the better start and were ahead inside five minutes.
Craig Gardner found Kemar Roofe with a throughball, and the young midfielder made no mistake in stroking a finish into the back of the net.
Port Vale goalkeeper Johnson was forced to tip over a Victor Anichebe effort 11 minutes later, but the Premier League side soon found themselves behind against the run of play.
Ben Williamson levelled with a half-volley from close range after Ryan Lloyd had headed down a cross from the right, and Tom Pope's glancing header from a Michael O'Connor free-kick gave the hosts an unlikely lead.
West Brom failed to make their possession count and were unable to cause Johnson too many problems.
However, Bradley Garmston charged into the penalty area four minutes after the break, before unleashing a fierce left-footed strike to pull the Premier League outfit level.
West Brom pressed for victory but were ultimately undone by Dawson's late lapse, although Gardner struck the post before full-time as the visitors came close to salvaging a draw.
After watching his new employers concede three goals to League One opposition, left-back Davidson may fancy his chances of being thrown into the side for Saturday's final pre-season friendly against Porto.