They will now play the winners of the fourth quarter-final, between the Blue Bulls and EP Kings on Saturday.
The Argentine Pampas team did well to recover from being 17-0 down in the first 10 minutes, after two tries by Specman and seven points from Bezuidenhout, to close the gap to 27-15 at half-time and to 34-30 midway through the second half.
Both teams scored five tries, but the Pampas were undone by some wayward kicking by their fly-half, Santiago Gonzalez Iglesias, who missed two conversions and three penalties.
Their scoreline was given some respectability by a try on the stroke of full-time after a series of concerted attacks on the Pumas line, all of which were held by a resolute defence until it was too late to make a difference.
The Pumas, who had lost just one game, heading the Northern section, showed they were prepared to be adventurous from the outset.
Full-back Coenie van Wyk released Specman 60m out with a well-timed pass in the fifth minute, then centre Stefan Watermeyer put through a perfectly-weighted grubber which sat up for Specman just two minutes later.
Along with the penalty slotted by Bezuidenhout, and two conversions, the Pumas were looking to be in total control, especially as the Argentineans were struggling in the scrums. But they hit back a try by flank Rodrigo Baez, converted by Iglesias, to narrow the gap to 17-7.
The two No 10s traded penalties, but the Pumas surged ahead again to 27-10 when Bezuidenhout kicked ahead for Specman. The bounce eluded the winger, but Bezuidenhout was on hand himself to gather and score.
Again the Pampas came back, with scrum-half Tomas Cubelli breaking free from broken play to score in the corner and they went into the break at 27-15.
The Argentines set the pace early in the second half, scoring from a line-out drive after three minutes, and adding a penalty, but the Pumas loosehead prop, Vincent Koch, playing wide, broke through the back line at speed for a converted try to push the gap to 34-23.
Again the Pampas responded; fullback Ramiro Moyano, benefiting from a wonderful offload by Cubelli. Then came Specman’s hat-trick try, with 10 minutes to play, and the game was out of reach.
Scorers:
Pumas 44 (27):
Tries: Rosko Specman (3), Carl Bezuidenhout, Vincent Koch
Conversions: Bezuidenhout (5).
Penalties: Bezuidenhout (3)
Pampas XV 37 (15):
Tries: Rodrigo Baez, Tomas Cubelli, Benjamin Macome, Ramiro Moyano, Manuel Montero;
Conversions: Santiago Gonzalez Iglesias (3)
Penalties: Iglesias (2)