Wales endured a tense Battle of Britain with Northern Ireland at the Parc des Princes but secured victory 15 minutes from time, when man-of-the-match Bale prompted an own goal from Gareth McAuley with a low, fizzing cross.
Chris Coleman and his squad have been joined by their families after five weeks in training camp and Bale celebrated on the pitch with his young daughter Violet Alba.
"To get through it with a win was the main goal. The feeling after was difficult to describe - very happy and very emotional," he said.
"To be able to share it with my daughter and my family was very emotional for me. As you say, I haven't seen them for four or five weeks.
"It was an amazing experience that I'll never forget. We have to keep playing and keep winning and we will keep trying to do that."
The Real Madrid forward's run of scoring in each match at Euro 2016 came to an end and he knew the free-flowing nature of the 3-0 win over Russia was unlikely to be repeated against Michael O'Neill's well-drilled side.
"We knew it was going to be a very difficult game, we knew how Northern Ireland play," he said.
"They are very defensive, shut up shop and make it difficult to score.
"We knew it would be hard for the attackers to get space but the team as a whole worked incredibly well.
"We had that one moment of quality and forced them into a mistake. That won us the game."
A quarter-final against Belgium - who Wales took four points off during qualification - or Hungary now lies in wait and Bale is determined to carry on his French adventure for as long as possible.
"It was similar in qualifying, you get one win and momentum grows and confidence grows," he added. "You ride the wave.
"We've had difficult games, in particular this one, but we've battled like we've always done. We've shown our team spirit and our passion.
"We need to keep riding this wave and we'll work hard every game to get as far as we can."