Wales fell 6-0 behind early on but from that moment looked the likely winners in the Cardiff contest as they showed considerably more control in possession and defensive solidity than in the 55-23 hammering they endured against New Zealand last weekend.
Argentina were coming off victory over England but were well below the standard they set in that game, with numerous unforced handling errors and poor kicks before prop Nahuel Tetaz Chaparro (33) scored their only try late on.
Wales host Georgia next Saturday, while Argentina travel to Murrayfield to play Scotland.
"We knew last week we'd have a lot of stick and rightly so and this just shows the character in the team to bounce back," Wales captain Justin Tipuric told Prime Video.
"We made it hard work in the end, we know we've got to sharpen up on a few things and put another performance in next week against Georgia.
"The boys stood up and put everything right. We could have come here today and let Argentina steamroller over us like last week, but we came fighting, met them and came away with a win."
A pair of Emiliano Boffelli penalties had Argentina ahead as Wales made a slow start to the game, but as the first half wore on it was the home side who applied the pressure.
Hooker Ken Owens had the ball ripped from his grasp as he fell over the line to score a try but moments later Wales got their opening score just past the half-hour mark.
They were rewarded for choosing an attacking line-out from a kickable penalty and comfortably mauled the ball over the line with Faletau dotting down on his 32nd birthday.
Flyhalf Gareth Anscombe added a penalty to give the hosts a 10-6 halftime lead.
That advantage was extended early in the second half when Williams charged down Argentina fullback Juan Cruz Mallia’s poor attempted clearance and had an easy chance to score unopposed.
Mallia is more used to playing on the wing than in the number 15 jersey and it showed.
Argentina had plenty of possession after that but were sloppy and failed to make it count, even after Wales lock Will Rowlands was sent to the sin-bin.
They got a try 12 minutes from the end as Tetaz Chaparro was at the back of a driving maul and dotted down, but the home side were able to comfortably see out the remainder of the game as they pinned Argentina in their own half.
Wales coach Wayne Pivac will have concerns over arm injuries to loose-forward Dan Lydiate and Rowlands, who were both forced to leave the field.
