Davies kicked five penalties, a drop goal and converted a try from Hallam Amos in a flawless kicking display, one which secured a narrow 25-20 win for Danny Wilson's side.
Wales had roared into a comfortable interval lead, Amos' superb individual try and 11 points from Davies giving them a 16-0 cushion.
But in an extremely physical second half, Argentina stepped up a gear, dominating possession and territory for long periods. A try from Pablo Matera got them back into things and with Patricio Fernandez finding his kicking range, Wales were up against it.
Crucially though, Wales toughed it out at the key moments and despite losing some men to injuries, they always kept Argentina at arm's reach. Fernandez had converted Matera's score and kicked a couple of penalties but Davies kept Wales ticking over nicely.
And despite 14-man Argentina crossing for a late try, Santiago Cordero the scorer, Wales had done enough to reach the last four.
With just over a minute on the clock in the first half, a penalty from Davies had given Wales a 3-0 lead and although Argentina responded with some concerted pressure, Ellis Jenkins' side held firm.
Argentina certainly boasted plenty of power upfront, the grunt of their forwards evident in the early exchanges. But with the Welsh pack more than a match for their opponents, the men in red soon came close to the first try of the day. Owen Jenkins broke clear and chipped ahead, the Pontypridd wing chasing his own kick as the ball rolled towards the try line.
Sadly for Wales, a score failed to materialise, Argentina full back Cordero covering impressively to clear the danger.
It wasn't long before Wales did score though and when the try arrived, it was certainly worth the wait. From a scrum inside the Wales half, Davies spread the ball left to full back Amos. The Dragons man cut a lovely line, leaving the Argentina defence for dead and showing an impressive turn of pace to finish from all of 50 metres. The touchline conversion was not a problem for Davies and midway through the first half, Wales led 13-0.
Things then got even better, Davies continuing his fine form with the boot with a snap drop goal. With little on at a ruck, Jenkins stepped in at scrum half and found the Ospreys man in the pocket. On his left foot, Davies did the rest, pushing his side further clear.
It was a lead they maintained until the interval, Argentina's only chance of points wasted when fly half Fernandez put a fairly simple penalty kick wide.
If Argentina needed a fast start to the second period, it failed to arrive immediately. Davies' growing confidence was shown through a short restart and a clean line break which nearly led to a second Welsh try. But a side of Argentina's quality was bound to wake up at some stage and they did just that, Matera burrowing over from close range. Fernandez converted and then kicked his first penalty of the day as the South Americans roared back into contention.
With Argentina boasting 70 per cent of the possession, Wales were up against it, something compounded by an injury to lock Rhodri Hughes. Fernandez then added a second penalty and with Wilson watching his side's lead evaporate, the Wales head coach rang the changes.
Jordan Williams and Elliot Dee were just two of the players introduced, a move which paid off immediately when a Davies penalty pushed Wales' lead back out to six.
Despite their advantage, Wales were being made to work hard for their points, the fact they had made 80 tackles compared to 31 from Argentina on the hour mark an indication of the amount of ball held by their opposition.
Up against it, Wales need a moment of magic and replacement Williams provided just that. The full back produced a superb step to escape the Argentina defence and offloaded to his namesake Rhodri. In plenty of space, the Llandovery scrum half was taken out by a high tackle from Guido Pagadizabal, an offence which earned the Argentina lock a yellow card.
Davies made Argentina pay from the resulting kick and although Cordero broke through to cross for a late converted try, a battered and bruised Wales held on with Davies applying a late coup de grâce.
Tries: Hallam Amos; Conversions: Sam Davies; Penalties: Davies (5); Drop Goals: Davies.
Tries: Pablo Matera, Santiago Cordero; Conversions: Patricio Fernandez (2); Penalties: Fernandez (2).