Belgium had several chances to go ahead in the first half before Radja Naiggolan crashed home a fierce long-range shot to take the lead.
Chris Coleman’s men came back strongly, however, and drew level when Ashley Williams headed in on the half-hour.
And in an open second half it was Robson Kanu who struck with a marvellous turn and finish before Sam Vokes added a third to earn Wales their first major semi-final in history.