Chris Coleman's men, who are competing in the nation's first major tournament since 1958, took the lead in the 10th minute through Bale's free kick.
They looked in trouble when Ondrej Duda scored an equaliser early in the second half, but Robson-Kanu came to the rescue off the substitutes' bench to give Wales three crucial points.
It could have been a nightmare start for Wales inside the first three minutes of the clash when Marek Hamsik began a charging run.
The Napoli midfielder stormed past three Wales players before cutting inside and unleashing a shot, but Ben Davies continued to track his opponent and managed to get in a vital block before the ball rippled the back of the net.
Joe Allen had an opportunity to get on the scoresheet when a weak punched clearance from goalkeeper Matus Kozacik fell to the feet of the Liverpool midfielder, but he failed to get a shot away and the ball was scrambled clear.
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Wales fans were in dreamland in the 10th minute of the match when their star player Bale smashed the ball into the back of the net from a free kick.
The Real Madrid attacker had some help from the keeper as the Slovakian took a step to his left just as Bale struck, but the 26-year-old did not care as he led wild celebrations with the players and coaching staff.
Slovakia struggled to get back into the match and despite goalkeeper Danny Ward making his international debut due to Wayne Hennessey's back injury, the Liverpool youngster was rarely tested in the first half.
It was worthy of a penalty and red card to Skrtel, but none of the officials flagged, and the Liverpool defender, who missed a chance towards the end of the half, escaped punishment.
It was a different story in the second half as Slovakia came out of the dressing room with intent, and Robert Mak in particular impressed with his movement.
The midfielder got in behind the defence and latched on to a low ball before firing a shot towards goal, but his attempt just fizzed over the crossbar.
Bale could have had a second goal when he got a header away after latching on to Allen's cross, but the ball bounced in front of the keeper, who prevented it from sneaking past the post.
The momentum shifted in the 61st minute when Duda scored with his first touch after coming off the substitutes' bench.
The midfielder broke away from his marker before drilling a low shot past Ward, and from that moment, Slovakia took control and peppered the goalkeeper with shots.
Ward stopped with his feet when Kucka fired an attempt towards goal, and the Liverpool keeper was called on again to punch clear a ball into the box.
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Aaron Ramsey missed a clear chance when he squandered a header from six yards out by sending it over the bar, but Wales were not finished as substitute Robson-Kanu fired the nation in front.
It was not the cleanest of strikes as he wrapped the ball around his left foot in the penalty area, but it bobbled into the back of the net.
Both sides had opportunities on the front foot, but Wales managed to hold on and seal a victory.