Bale was kept on a tight leash by Northern Ireland in a cagey Battle of Britain in Paris, but he was instrumental in the winning goal when Gareth McAuley diverted the forward’s 75th-minute cross into his own net.
Wales will now play Belgium in the quarter-final on Friday, and the Real Madrid star believes they have got momentum to hopefully go even further in the tournament.
“It’s similar to qualifying, you get one win and the confidence grows,” said Bale.
“We’re very confident at the moment, we know we’ve had difficult games.
“But we’ve battled today, shown our team spirit and passion and we just want to carry on riding this wave.
“We will keep working hard to get as far as we can.”
Bale celebrated victory at the final whistle with his three-year-old daughter Alba Violet.
“The feeling after was difficult to describe, very happy, very emotional,” said Bale.
“To be able to share it with my daughter and my family was emotional for me.
“I haven’t seen them for four or five weeks.
“It was an amazing experience that I will never forget.
“But our ambition is to keep fighting on and win every game we play.
“We knew it would be difficult today, but we had that one moment of quality that forced them into a mistake and ultimately that won us the game.”
Bale eventually left his mark on proceedings deep into the second half.
It came as Northern Ireland, who had fought and harried for their lives, were within 15 minutes of taking the game to extra time, but punished for switching off for a split second.
In that time Aaron Ramsey found Bale in space he had craved all day, and he made the most of it with a perfect ball into the box between McAuley and goalkeeper Michael McGovern.
McAuley had to intervene with the substitute Hal Robson-Kanu lurking behind, but his touch from a few yards out was a fatal one for Northern Ireland.
Wales might have come into the game as favourites after winning their group and having the tournament’s joint top scorer in Bale, while Northern Ireland qualified for the last 16 as one of the four best third-placed teams.
But Northern Ireland can boast a FIFA world ranking of 25, one lower than Wales, and their run of only two defeats in 15 games –albeit to Germany and Poland at this tournament – demanded respect.
“We didn’t really play well at all, but that wasn’t our fault,” said Wales manager Chris Coleman.
“Credit to Northern Ireland who did what they do well, and that’s make it very difficult for the opponents. It was an ugly win – but who cares? We’ve sat here before and played very well and not got a result.
“It’s a good sign because in the three wins out of four we’ve had to win in different ways.
“I thought it would be difficult but they were tougher than I expected.
“If I’m honest we would have been better if we’d played Turkey.
“It’s not because Turkey are not a good team, because they are, but it’s different when you play a home nation.
“I was always a bit nervous coming into this game, but we tried to work on not getting too emotional playing a neighbour.
“We played so well against Russia, which was a complete performance, but there was so much riding on this game. It was going to take a little bit of luck and we got it with the goal we scored.”
The closest Northern Ireland came was as early as the 10th-minute when Leeds United’s Stuart Dallas forced a fine save from Wayne Hennessey.
For Michael O’Neill, for their superb travelling support and for the passionate McAuley it was a bitter denouement to the nation’s first major tournament in 30 years.
“I felt it was a very tight game. It’s just a very, very cruel way to lose the game,” he said.
“We didn’t deserve to lose the game...certainly with that nature (of goal).
“Not many people gave us any chance of doing anything in the tournament but we came within a whisker of getting to the last eight. “I don’t think it would have been undeserved had we got that.
“It’s difficult to quantify and reflect on the tournament as a whole at this minute, but the players have been tremendous in their effort, their commitment, their pride in playing for their country.
“It’s been a fantastic experience for everyone but when it ends the way it did today it’s very cruel. I don’t think we got what we deserved.”
Wales: Hennessey, Chester, A Williams, Davies, Gunter, Allen, Ledley (J Williams 62), Ramsey, Taylor, Vokes (Robson-Kanu 55), Bale. Unused substitutes: F Williams, King, G Williams, Edwards, Richards,Cotterill, Collins, Vaughan, Church, Ward.
Northern Ireland: McGovern, Hughes, McAuley (Magennis 84), Cathcart, J Evans, Ward (Washington 69), Davis, C Evans, Norwood (McGinn 79), Dallas, Lafferty. Unused substitutes: Carroll, McLaughlin, Ferguson, Baird, Grigg, McCullough, McNair, Hodson, Mannus.
Referee: M Atkinson (England).