Robson-Kanu fired home from inside the area in the 81st minute following a neat pass by Aaron Ramsey to secure the three points after Bale had put the Welsh in front in the 10th with a free kick that deceived Slovakia keeper Matus Kozacik.
Slovakia, who should have scored after five minutes when Marek Hamsik dribbled past three players only to see his goalbound shot cleared off the line by Ben Davies, equalised through substitute Ondrej Duda with his first touch on the hour.
But Robson-Kanu popped up to give Wales a deserved victory in their first major international tournament for 58 years after an inspired substitution by manager Chris Coleman who was as delighted as the hordes of Welsh fans at the Stade de Bordeaux.
