The result left Arsenal in fifth place with 50 points from 27 matches, five behind fourth-placed Liverpool who are at home to Manchester City on Sunday. West Brom stayed eighth on 43 points from 29 games.
Still reeling from a 10-2 aggregate defeat by Bayern Munich in the Champions League last 16 earlier this month, the under-fire Wenger could only watch in dismay as a banner saying 'NO CONTRACT #WENGEROUT' was flown by a plane above the Hawthorns.
He was just as helpless in watching another dismal performance from his side, who conceded two goals from set pieces as sloppy defending cost them a chance to close in on the top four.
Dawson fired the home side ahead in the 12th minute after a Nacer Chadli corner before Alexis Sanchez gave Arsenal a ray of hope three minutes later with a composed finish from a superbly weighed pass by midfielder Granit Xhaka.
The opening half produced relentless pace and a string of chances at either end, as both goalkeepers were called into action several times.
Arsenal's Peter Cech limped off with a muscle injury in the closing stages of the first half and that proved costly as his replacement David Ospina was at fault for West Brom's second goal.
Robson-Kanu pounced on a mix-up between Ospina and an Arsenal defender 75 seconds after coming on as a substitute for Salomon Rondon to poke the ball into an empty net in the 55th minute.
Desperate defending then denied Robson-Kanu and Chadli as Arsenal looked all at sea whenever the home side came forward, while Danny Welbeck saw his header cannon off the crossbar at the other end.
More poor marking, with Arsenal players positioned like statues for a James McLean corner, allowed the towering Dawson to seal West Brom's win with a thumping close-range header in the 75th minute.
(Writing by Zoran Milosavljevic; Editing by Clare Fallon)
