The Baggies went into the game on the back of successive defeats to Crystal Palace and Everton while Arsenal had lost three of their last five games across all competitions.
Both teams were in desperate need of three points and it was the hosts who caused a huge shock at the Hawthorns.
A brace from defender Craig Dawson and a close-range strike from Robson-Kanu were enough to see off a struggling Gunners side, who only had an Alexis Sanchez goal to show for their efforts.
Tony Pulis' men remain eighth in the table, seven points behind both Arsenal and Everton, who are sixth and seventh respectively, and the West Midlands club still have a slim chance of finishing in a Europa League position.
Robson-Kanu, who has been used sparingly this season after joining last summer, wants his side to use the win to finish their campaign on a high.
He told the club's website: "We're certainly not taking our foot off the pedal. The manager has said he wants us to work even harder now we're at that 40-point mark.
"We're back to winning ways and hopefully we can build some momentum and kick on again."