After losing to South Africa by a point in their opening Pool D game, Wales knew defeat against opponents that had beaten them twice in previous World Cups would effectively mean an early flight home.
But Jones and company dug deep, securing a 17-10 triumph and setting the tone for successive victories over Namibia, Fiji and Ireland to set up next Saturday's semi-final showdown with France in Auckland.
"We took a lot of confidence out of the South Africa game," Jones said, reflecting on a 17-16 loss.
"We thought we could have won it, and then I think the big turning point was when we beat Samoa.
"We knew Samoa would be a tough side, but we came through it and we are now ticking over nicely.
"We were in a tough group - Samoa and Fiji had been bogey sides for us - but we knew if we came through we could have a good crack against anyone."
Ospreys prop Jones was part of a Wales World Cup squad that failed miserably in France four years ago, being dumped out by Fiji before the knockout phase and resulting in coach Gareth Jenkins losing his job.
And the contrast between those numbing events in Nantes and Wellington this weekend could hardly be greater, with Wales now standing one win away from a first World Cup final appearance.
"It's certainly a better feeling than four years ago," Jones added. "There is a different atmosphere in the changing rooms, too.
"Results-wise, a few of the journalists back home in Wales are going to struggle to write bad things about us now.
"The boys are really enjoying their rugby at the moment, enjoying the ride, so to speak. We just want to play rugby without any baggage.
"I thought our back-line was fantastic against Ireland, and it was a case of a few of us older guys keeping up with the younger boys in the team.
"These young boys in the squad are so fit and professional that people like myself and Gethin (fellow prop Gethin Jenkins) have got to try to keep up with them. If you don't, you fall by the wayside.
"A lot of the older boys were pretty shattered afterwards, but all the younger boys were jumping around dancing. You don't get to many semi-finals, after all.
"It's a push on now to the finals, and we are just going to enjoy it."
Jones also reserved praise for Wales' 23-year-old captain Sam Warburton - the youngest skipper in Rugby World Cup history - and the part he has played behind a stunning resurgence.
"What is good about Sam is that he is not one of these ranters," Jones said.
"He doesn't speak for the sake of it. He doesn't say a lot, but what he does say makes sense. He leads by example.
"He is fantastic, really. He is right up there at the moment, and there is a belief that has developed within the squad."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The Barbarians are one of the top invitational rugby sides with a long history and classic rugby jersey.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Lions get ready for their matches with this green training jersey. It's what the players wear. Get in the Gear!
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.