Chris Coleman's men saw Group D rivals Republic of Ireland drop points before they kicked off at the Principality Stadium, but they struggled to find a breakthrough against the visitors for much of the match.
"It was a make-or-break game today and we have had it in the past and we have stood up to the challenge and the most important thing was the three points today," Bale told reporters.
"We always speak about the performance but tonight was just great determination. In the first half, we were a little bit off the pace and we had a good chat at half time and just calmed it down.
"The manager has always said it is a squad game and [Woodburn]'s come and a what a great debut, I'm so happy for him."
Wales are next in action away to Moldova on Tuesday.