Arsenal fell behind to Dawson heading in Nacer Chadli’s corner on 12 minutes.
Alexis Sanchez equalised three minutes later but Arsenal saw their goalkeeper Petr Cech go off injured.
Albion restored their lead on 55 minutes when Hal Robson-Kanu prodded in a minute after coming on and, after Danny Welbeck had struck the bar for Arsenal, Dawson struck again with a header, on 75 minutes, from James McClean's corner.
West Brom stay eighth, on 43 points, while Arsenal are five points off fourth-placed Liverpool after a fourth defeat in five league matches.
Man of the Match: Craig Dawson (West Brom)