Friday’s contest marked Wales’ first quarter-final in 58 years, their other run to the last eight coming in the country’s only previous appearance at a senior international competition, the 1958 World Cup in Sweden.
But just 13 minutes in Chris Coleman’s side were given a mountain to climb against the world’s second-ranked nation as a fierce long-range strike from Radja Nainggolan put Belgium ahead.
Wales’ reply was magnificent and a deserved leveller came courtesy of captain Ashley Williams, who capitalised on some poor defending to restore parity.
Belgium appeared set to take command of an enthralling contest once more after starting the second half on the front foot, only for Robson-Kanu to evade three players with a marvellous turn and make it 2-1 in confident fashion.
Aaron Ramsey was then booked – meaning the instrumental Arsenal midfielder will miss the semi-final against Portugal, which Vokes made absolutely sure of progressing to with a fantastic near-post header from Chris Gunter’s cross.
Only some heroic defending prevented Wales from falling behind in the seventh minute as Belgium went agonisingly close to opening the scoring on three occasions.
First Yannick Carrasco was thwarted by a fine close-range save from Wayne Hennessey following excellent work down the right from Kevin De Bruyne, with Thomas Meunier’s follow-up blocked by Neil Taylor before Eden Hazard saw his effort deflected over.
Romelu Lukaku was unable to connect with the subsequent corner from De Bruyne and Belgium were fortunate not to see his profligacy punished as Robson-Kanu headed over a Taylor cross and Gareth Bale lashed into the side-netting following a trademark surge down the left.
But Belgium were rewarded for a strong start in remarkable fashion through Nainggolan.
The Roma midfielder was afforded too much space as he picked the ball up 30 yards from goal and unleashed a vicious strike that Hennessey could not keep out, despite getting a hand to it.
Wales responded well and Belgium were grateful to the reflexes of Thibaut Courtois for preserving the lead, the Chelsea goalkeeper somehow keeping out Taylor’s low drive after the left-back had been picked out in the box by Ramsey.
Belgium did not pay for their poor marking on that occasion, but Williams took advantage of more questionable defending on the half-hour mark.
The skipper was left unmarked and met Ramsey’s corner with a downward header into the bottom-right corner.
Belgium began the second 45 in the ascendancy as Lukaku missed a gilt-edged chance by heading a Meunier cross wide and De Bruyne and Hazard both went close with curling efforts.
However, Wales completed the turnaround 10 minutes after the restart, Robson-Kanu displaying excellent composure to fire them in front.
He latched onto a superb left-wing cross from Ramsey and then produced a moment of magic before coolly slotting beyond Courtois.
Marouane Fellaini headed wide from the edge of the six-yard box and Nainggolan saw a penalty claim waved away, but it was Vokes who had the final say five minutes from time to stunningly knock out one of the heavy pre-tournament favourites.