Ten minutes into the second half, with the game poised at 1-1, Robson-Kanu produced a perfect Cruyff turn to leave two Belgian defenders for dead before slotting past Thibaut Courtois.
Sam Vokes then finished off the win five minutes from time as Wales moved into the last four in their debut European Championships.
Aaron Ramsey had set up Robson-Kanu's score, just as he had Ashley Williams for a first-half equaliser after Radja Nainggolan's opener, but the Arsenal man will miss the semi-final after picking up a second yellow card of the tournament.
Belgium had made an ominous start with Neil Taylor having to make a brilliant block on the line from Thomas Meunier in the middle of a flurry of shots after Kevin de Bruyne had released Romelu Lukaku.
Wales were on the ropes, with Ben Davies also ruled out of the semi-final with a booking, and they went one down after 13 minutes following a Nainggolan thunderbolt. With Wales sitting off the midfielder, the space opened up and Nainggolan needed no second invitation, hammering a first-time shot into the top corner past the despairing Wayne Hennessey from 25 yards.
Under the cosh, Wales responded well and almost drew level from Taylor after Ramsey had found space on the right. Courtois made a fine save, but Belgium clearly didnt learn their lesson and soon after Wales equalised.
On the half-hour Williams was left completely unmarked from a corner and made Belgium pay, powering a header past De Bruyne, who had abandoned his position on the near post.
He should have had a second before the break but headed over when left unmarked again as Wales finished the half in the ascendancy.
The break seemed to do Belgium good though and Lukaku should have put them back in front when he leaped highest to a Meunier cross only to glance wide.
But it was Wales who took the lead thanks to a moment of magic from Robson-Kanu. Ramsey got away down the right again, found the striker who produced a brilliant Cruyff turn to find himself one on one with Courtois and fired home with his left foot.
Belgium were rattled, and when Courtois fumbled at a corner soon after Williams had a chance to extend the lead but couldn't get the ball past the Belgian keeper.
Still in the last 20 minutes momentum started to shift and half-time substitute Marouane Fellaini should have equalised from Toby Alderweireld's brilliant cross, but somehow headed wide from six yards out.
Davies then got lucky when he appeared to trip Lukaku for what would have been a second yellow card, but wasn't penalised by Damir Skomina, who also let off Williams when he caught Nainggolan in the box.
Instead it was Wales who made the game safe with five minutes remaining when Vokes beat Alderweireld to Chris Gunter's cross and headed past Courtois.
STATS
Belgium 1
Nainggolan 13
Wales 3
A Williams 30, Robson-Kanu 55, Vokes 85
Referee: Damir Skomina 5/10
Match rating: 4/5
Attendance: 45,936
Star man: Aaron Ramsey (Wales)
Belgium: (4-5-1): Courtois 6, Meunier 6, Alderweireld 5, Denayer 4, J Lukaku 4 (Mertens 75 5), Nainggolan 7, Witsel 6, Carrasco 6 (Fellaini 45 4), De Bruyne 5, Hazard 5, R Lukaku 5 (Batshuayi 84 5)
Wales: (5-3-2): Hennessey 7, Gunter 7, Chester 7, A Williams 8, Davies 7, Taylor 7, Allen 8, Ledley 7 (King 78 6), Ramsey 8 (Collins 90 N/A), Robson-Kanu 8 (Vokes 80 7), Bale 7
Sportsbeat 2016
Source: Sportsbeat
