Wales came agonizingly close to beating the Springboks when they faced the World Champions in Cardiff last weekend but Byrne insists that finishing as gallant losers is no longer enough for Warren Gatland's men.
"Warren gives us a lot of self-belief. We can go out there and win - I honestly believe that," said Byrne ahead of the first Test in Dunedin.
"We know we can compete - we have shown that in the last couple of years. But it's a question of finishing sides off.
"We have to start being as clinical as the best teams in the world. When they put themselves in good positions, they come away with victories. Winning is the most important thing."
Byrne's comments are perhaps somewhat unsurprising when you consider the nature of Wales' most-recent defeat.
Wales played brilliantly in stages but, despite having opened up a 16-3 lead over the Boks at the Millennium Stadium, even a late comeback couldn't make up for a disappointing middle period of the game as Wales lost out 34-31.
"It's not often that you get a chance to beat the best sides in the southern hemisphere, so of course we are disappointed," added Byrne.
"But in a weird way it's good that we're unhappy about losing to the World Champions. It shows that we are setting high standards."
Those high standards will have to be stepped up another level if Wales are to achieve something they've never achieved before either next weekend or the week after in Hamilton.
Wales have never beaten the All Blacks on New Zealand soil and Byrne is fully aware that Wales will have to be at their very best to have a chance of breaking that duck.
"They've lost a lot of players to overseas, but any New Zealand team will be up there with the best in the world.
"You give them a sniff and they'll take a try. We will have to be at the top of our game, otherwise there could be big scorelines."