The home side started brightly and were rewarded in the 12th minute when Dawson headed home Nacer Chadli's corner thanks to some poor Gunners' defending.
There was an immediate response from the visitors, however, when Alexis Sanchez found space in the box and slammed home his 18th Premier League goal of the season.
The visitors lost goalkeeper Petr Cech to injury in the 38th minute but the sides went into the break all square.
However, the Baggies re-took the lead just 10 minutes into the second period as substitute Hal-Robson-Kanu made an immediate impact, sliding home with just his second touch.
The points were sealed fifteen minutes from time when Dawson headed home his second goal of the game, also from a corner.
Wenger is under pressure like never before at the Emirates, with loud boos and banners declaring 'enough is enough' at the final whistle hardly likely to lighten the Frenchman's mood.