Schwartzman claimed the third victory of his career over a top-five opponent, winning 6-7 (5-7) 6-3 6-3 in Sydney.
The world No.13 fought back from a break down in the second set before taking his career head-to-head with Tsitsipas to 2-1 after battling through two hours and 42 minutes at Ken Rosewall Arena.
"I was just trying to think that it was his first match for two months after his elbow [injury], so I was thinking just to try and be in the match," Schwartzman said after the match.
"I had the chance and I think I was ready. I am very happy because playing against Stefanos and being a set and a break down is not easy, but I found a way."
Argentina, which earlier saw Federico Delbonis defeat Michail Pervolarakis in straight sets, will now face Poland on Wednesday to decide who progresses to the semi-finals.
Poland was a comfortable 3-0 winner in its match-up with Georgia, Kamil Majchrzak beating Aleksandre Bakshi for the loss of just two games before Hubert Hurkacz defeated Aleksandre Metreveli 6-7 (5-7) 6-3 6-1.
Pablo Carreno Busta beat Viktor Durasovic 6-3 6-3 to put Spain ahead against Norway, and its win was assured when Roberto Bautista Agut claimed an impressive 6-4 7-6 (7-4) victory over Casper Ruud. Alejandro Davidovich Fokina and Pedro Martinez followed up with a straight-sets doubles win.
"Casper is playing unbelievable tennis, [he did] an unbelievable performance last year, and today I played very good," Bautista Agut said. "I returned very well, I made very few unforced errors and I played aggressively. I try to play matches like this and that is why I am practising hard and I am trying to show this level."
Spain is top of Group A ahead of Serbia, whose match with Chile will go down to a doubles decider.