Playing their first game at a major championship since the 1958 World Cup, Wales took a 10th-minute lead courtesy of Gareth Bale's fine free-kick, only for Slovakia substitute Ondrej Duda to equalise yesterday.
But with nine minutes remaining, Robson-Kanu, released by Reading at the end of the season, scuffed a shot past Matus Kozacik to send Chris Coleman's side striding towards the last 16.
His strike ended Wales's four-game winless run and tees the European Championship debutants up perfectly for Thursday's hotly anticipated second Group B fixture against neighbours England in Lens.
Jan Kozak's Slovakia, previously unbeaten in eight games, tackle Russia in Lille on Wednesday knowing that a second defeat could be fatal.
The match at the elegant Stadium Bordeaux had been billed in some quarters as a battle between Real Madrid star Bale and Marek Hamsik, and the Slovakia playmaker was the first of the pair to catch the eye.
After robbing Bale, he surged past Ashley Williams, David Edwards and James Chester before drilling a left-foot shot past goalkeeper Danny Ward, making his first start in place of the injured Wayne Hennessey, but Ben Davies slid in almost miraculously to divert the ball behind for a corner.
When Bale's moment arrived 10 minutes in, there was to be no such reprieve for Slovakia. Presented with a free-kick 30 yards from goal, he slammed the ball over the wall and as Kozacik moved to his left, it darted in the other direction and plunged low into the net.
Bale tore down the touchline in celebration of his 20th Wales goal, which came almost 10 years after he opened his international account with a free-kick against the same opposition.
"Are you watching England?" crowed the Wales fans, before honouring former manager Gary Speed, who took his own life in November 2011, with a cry of "There's only Gary Speed!"
Boos emanated from the red-clad hordes in the Wales end moments later, however, as a replay on the stadium's big screens revealed that Martin Skrtel had escaped punishment after catching Jonny Williams in the throat with his elbow, barely six feet from one of the goal-line officials.
The teams exchanged chances early in the second half -- Robert Mak drilling over, Bale sending Kozacik scrambling with a downward header -- before a double substitution brought Slovakia back into the game.
Slovakia coach Kozak sent on Adam Nemec and Duda, with Michal Duris and Patrik Hrovosky giving way, and within a minute Duda had scored, arrowing a shot past Ward's left hand from Mak's right-wing cut-back.
It made him the first player to score past Wales at a major tournament since a 17-year-old Pele at the 1958 World Cup.
Hamsik and Mak threatened before Coleman made a double change of his own, sending on Joe Ledley, 35 days after he fractured a fibula, and Robson-Kanu, who was to make a major impact.
After crossing for Aaron Ramsey to head over, the roles were reversed to decisive effect, the Arsenal man skipping past Skrtel and Robson-Kanu steering the ball home.