Wales came agonizingly close to emulating the Lions' third-Test victory over 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 a week on Saturday.
"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.
Having opened up a 16-3 lead over the Boks at the Millennium Stadium, Wales conceded 18 unanswered points to trail 21-16 early in the second half.
Things got even worse for Wales as the game progressed and even a late comeback couldn't prevent them from letting what would have been only their second-ever win over South Africa slip from their grasp.
"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 a 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, despite the hosts missing a number of high-profile stars, Byrne is fully aware that Wales will have to be at their very best just to stay competitive.
"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."