Half-back partners Rob Howley and Neil Jenkins were the driving force behind the success and nearly two decades on, both men are back to take on the Pumas, this time as coaches alongside Wales boss Warren Gatland.
In that time, Argentinean rugby has come on leaps and bounds.
The South Americans – as they always have and always will be – remain tough, uncompromising and with a fearsome set-piece. They are though, as Howley and Jenkins both acknowledge, far from a one-tricky pony.
“In 1999 we were the first northern hemisphere side to win over in Argentina, we were working under Sir Graham Henry then,” Wales attack coach Howley said.
“We played in Buenos Aires and they certainly asked questions of our strength in depth. Argentina were a very good team at the time, very strong up front, and they had Agustin Pichot at scrum-half.
“It was a memorable win for us, historic, and hopefully we can replicate that.
“We then went on to beat them at the World Cup and I’m really looking forward to meeting Argentina. There is added pressure on the players to keep producing the goods with some of the senior guys absent.”
Wales sealed a 36-26 victory over Argentina in the first Test in Buenos Aires in 1999 with the metronomic Jenkins kicking three conversions and four penalties.
A week later, the pair were at it again, Howley’s leadership at the base saw him get the better of Pichot as he cajoled his men to a 23-16 success.
Jenkins excelled again with five penalties and a drop goal. “From my memory I always remember Argentina being incredibly tough. They’re physical, uncompromising, and run hard to win the collisions,” the former fly-half said.
“The venues there are more football orientated, but the atmosphere is fantastic and it’s an amazing place to play.
“It was a long time ago now, but I enjoyed it when I played there so from my experience the rugby will be very tough. Argentina are going to be physical, there is no doubt about that.
“I’ve watched the Jaguares in Super Rugby this season against the Blues and the Chiefs and to go to New Zealand and win those two games was a very good achievement.
“If you look at Argentina over the last few years they’ve come into the Rugby Championship and now they’ve got a side in Super Rugby too. They’ve tried to evolve their game, they’re playing a lot more with ball in hand, and they’re showing off their skills as well too.
“They’re going to be very dangerous, but the boys will enjoy that I’m sure.”
Wales have won five of their last six meetings against Argentina, all of which have come in Cardiff, but the Pumas are formidable on home soil with their giant forwards and enterprising backs.
Gatland’s side arrived in South America on Monday fresh from their dramatic 22-20 victory over South Africa in the American capital of Washington.
“Argentina will be tough as they always are and you have to go back a long way since we’ve won a tour there,” said Ellis Jenkins, Wales co-captain this summer alongside Cory Hill.
“They’ve stepped up the world rankings a lot in the last 10 years or so. They’ll be tough, especially in their own backyard and the majority of their team have been playing together for a while now.
“It’ll be tough for us and that’s why we’ve spoken about having a limited preparation time and making the most of every session we have. It’s certainly going to be different.
“I’ve never been to Argentina, all I’ve heard is the steak is decent down there!”
The Buenos Aires-based Jaguares are flying high in Super Rugby and Pumas boss Daniel Hourcade is certainly confident, claiming his men are in “unbeatable” condition.
“With the way the Jaguares are playing at the moment, Argentina are going to be very dangerous,” Howley said.
“They’ll be smarting from the disappointment of their last home series against England and we’ll be ready for a response from a side that’s going the right way.
“With a World Cup around the corner, both teams will be desperate to impress.”