After two passionately-sung national anthems, both teams set about trying to gain an early advantage, and Ukraine's Oleksandr Zinchenko saw a long-range free kick cancelled out by the referee, who deemed that it had been taken too quickly.
Wales were having their own moments in attack, but it was the visitors who were creating the better opportunities, Wayne Hennessey being called upon several times during the opening 45 minutes.
As well as a number of interventions from shots from distance, the veteran stopper also had to get down to his right to deny Roman Yaremchuk after he had broke clear of the defence.
Nevertheless, with 11 minutes of the first half remaining, Wales went in front. The set piece from 30 yards was far from ideal for a left-footer - even Bale's - but the legendary forward's effort was deflected into the net by the head of Yarmolenko.
Moments after the restart, the Dragons should have doubled their lead, the initial brilliant work being done by Neco Williams and Bale before Aaron Ramsey directed a first-time effort from 10 yards wide of the far post.