Mayer won the second rubber 7-6(5) 7-6(1) 4-6 6-3 after a gruelling four-hour tussle in which he took a timeout to tend to a thigh problem.
World number 15 David Goffin gave Belgium a good start in their bid to reach the final for the first time in over a century when he beat Federico Delbonis 7-5 7-6(3) 6-3.
The winners will face Australia or Britain in the final which will be held in Belgium.
Darcis fought hard during a gruelling first set until Mayer pushed it to a tiebreak after 74 minutes on court.
The second set saw Darcis keep up the fight, although Mayer sailed through the second tiebreak.
The Belgian battled back in the third after losing the first game to lead 5-1 only to falter as Mayer clawed his way back to 5-4 before Darcis clinched the set 6-4.
Mayer went ahead in the fourth and, despite a time out to treat his leg problem, he held off a resurgent Darcis at 5-3 to claim the set.
“We’re still in it and we will be here on Sunday,” said Darcis. “It’s not catastrophic, it’s a shame because there would have been a way.”
Argentina are playing in their 10th semi-final in 14 years, reaching the final three times in that period, but have lost all of their four finals including their 1981 appearance.
Belgium, who are looking to reach the final for only the second time, the last being in 1904, are pinning their hopes on Goffin picking up a second singles win on Sunday.
Delbonis proved a tricky opponent for Goffin and broke back in the opening set after falling behind, only to falter at 5-5.
Goffin also broke in the second set but was again pegged back and needed a tiebreak to take control before gliding through the third set.
“It was a tough one but the atmosphere is always amazing here in Brussels,” Goffin, who enjoyed noisy support, said.
“Always when I was a break up he was playing so good and was really aggressive so it was always tough to finish the sets but in the end I’m really happy with the way I won today.
“The pressure is on their shoulders now.”
(Reporting by Julia Fioretti; editing by Martyn Herman and Ken Ferris)