Byrne and company booked a quarter-final appointment with Ireland in Wellington next Saturday after crushing Pool D opponents Fiji 66-0 at Waikato Stadium.
Wales' best World Cup performance was a third-place finish at the inaugural competition in New Zealand 24 years ago, but full-back Byrne and company are determined to go further than those 1987 exploits.
Asked if Wales could win the tournament, he said: "I would probably say so.
"We know Ireland played very well to beat Australia, and we will give them respect, but we will give it our best to try to go as far or further than any Welsh team has done in the World Cup.
"The hard work has been done. We're in great shape for the knockout rounds after coming through a difficult pool.
"We are here for another week, but we want to stay for longer. The boys are really enjoying being at the tournament and we have got high ambitions.
"We are confident about our prospects against Ireland. It's just a case of seeing what happens on the day."
Wales' young guns once again came up trumps, all playing key roles against Fiji, with skipper Sam Warburton, wings George North and Leigh Halfpenny, centre Scott Williams and replacement scrum-half Lloyd Williams - they have a combined age of just 104 - scoring tries.
Wales not only avenged their World Cup elimination defeat against Fiji in Nantes four years ago, but also inflicted a tournament record defeat on the South Sea Islanders.
"George has now scored nine tries in 11 internationals, which says it all," Byrne added. "He has got something special.
"He's big, physical and is also exceptionally quick. His passing and skills are good as well, so he's got it all really.
"George has played more games for Wales than the Scarlets, which is incredible when you think about it.
"I enjoyed the win and the points we scored, but to stop Fiji from getting anything was more pleasing. Our defence coach Shaun Edwards was delighted with the shut-out.
"We weren't really too worried about Fiji. We knew our fitness would tell in the end and we could rely on it.
"From the start we were fitter than Fiji. We fronted up to them in the forwards and they didn't like it, spilling the ball a bit.
"We knew how much they liked to run the ball, so to keep them to zero has really sent a message out to the rest of the world about the strength of our defence.
"I am sure Ireland would have been watching us, and we will see them in the quarters."
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