Dawson was left unmarked to head home from in-swinging corners in each half, condemning the Gunners to a fourth defeat in their past five Premier League outings.
Alexis Sanchez fired a 15th-minute equaliser - his 22nd goal in all competitions this season - but Albion substitute Hal Robson-Kanu, released by the Gunners as a schoolboy, restored the lead 10 minutes after half-time and just 75 seconds after coming on, as Arsenal protested in vain for offside against James McClean.
David Ospina, on in goal after a first-half injury to Petr Cech, was to blame irrespective of the visitors' complaints and it capped a miserable afternoon for Wenger, after the opening was played out amid a bizarre backdrop of light aircraft slogans.
A plane sporting a banner urging Wenger to stand down emerged at kick-off, although another trailing the slogan 'In Arsene we trust' appeared overhead around the time Sanchez restored parity.
The afternoon would get no better for Arsenal's 67-year-old manager, whose prospects of UEFA Champions League qualification are in grave danger.
Liverpool are five points better off in fourth, while sixth-placed Manchester United can leapfrog them with victory at Middlesbrough later this weekend.
The expected pattern of Arsenal dominating possession was established early but Tony Pulis's men hit the front in the 12th minute.
Cech beat a firmly-hit shot from McClean behind but Dawson was allowed a free run at Nacer Chadli's delivery from the left to glance home.
However, the two players involved in the Albion goal were culpable when Arsenal pulled swiftly level.
Right-back Dawson was drawn into the area, leaving space for Granit Xhaka to pick out Sanchez behind Chadli with a lofted pass and the Chile star chested down to smash in via the underside of the crossbar from eight yards.
Salomon Rondon, without a goal since a hat-trick against Swansea City in December, dragged wide as he sought to restore West Brom's lead, while Ben Foster saved superbly to deny Aaron Ramsey and Allan Nyom did enough to stop Theo Walcott from converting the rebound.
Foster's opposite number Cech was similarly impressive in thwarting Darren Fletcher in the 34th minute but the veteran goalkeeper limped out of the action before half-time due to an apparent calf problem.
Rondon got above Laurent Koscielny to glance wide from Chris Brunt's cross in the 51st minute, shortly before he made way for Robson-Kanu to make an instant impact.
McClean looked to get on the end of Chadli's lofted pass and was beaten to it by Ospina, but the Arsenal keeper inexplicably neglected to use his hands and bundled the ball into Robson-Kanu's path.
The Wales striker prodded into an unguarded net, with McClean stepping over the ball in an offside position to let it run in.
Danny Welbeck headed a 65th-minute corner against the bar for Arsenal before West Brom broke and Ospina's inauspicious outing continued.
The Colombia international was hesitant in coming from his line to block Robson-Kanu and fell face first as Chadli gathered the rebound, grateful to look up and see Shkodran Mustafi block on the line.
But Ospina would be dreadfully exposed like Cech before him when McClean sent in a right-wing corner with 15 minutes to play - Dawson nodding in and colliding with the similarly unmarked Jonny Evans before wheeling away in celebration.
'Wenger Out' banners of a more conventional kind then became increasingly visible as an entirely different mood gripped the away end.
Key Opta stats:
- Arsenal have lost four out of five Premier League games for the first time since April 1995.
- West Brom have scored a league-high 14 Premier League goals from corners this season.
- Craig Dawson has scored three goals in his last four Premier League appearances, as many as in his previous 55.
- Alexis Sanchez has been involved in 27 PL goals this season (18 goals, nine assists) – more than any other player in the competition.
- Overall, Sanchez has now been involved in 100 goals for the Gunners in all competitions, scoring 64 and assisting 36.