Shane Long had headed his first of the season after 41 minutes to give Southampton a deserved lead but Matt Phillips curled in an equaliser two minutes later.
Robson-Kanu, whose last goal was his Cruyff-turn classic for Wales in their Euro 2016 quarter-final victory over Belgium, had been on the bench since joining West Brom in August but delivered an impressive all-round performance capped by his powerful 20-metre strike.
Southampton, hammered 4-1 at home by Tottenham Hotspur last time out, did little to get back into the game and had centre back Virgil Van Dijk sent off for a second yellow two minutes from time.
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Southampton had started the day in eighth, one place and one point above West Brom, but the result means the teams swapped positions.
(Reporting by Mitch Phillips, editing by Ed Osmond)
