Higuain's early goal against Belgium took Argentina into the 2014 World Cup semi-finals where they meet the Netherlands, who came through a shootout with Costa Rica.
Substitute goalkeeper Tim Krul produced two decisive saves in the penalty shootout to lift the Netherlands to 4-3 over Costa Rica after a scoreless 120-minute play in the last quarter-final match of the World Cup in Salvador, reported Xinhua.
In an earlier match at Brasilia, Argentina reached the last four for the first time since 1990 by beating Belgium 1-0 through a goal from Gonzalo Higuain, his first in this World Cup.
The 26-year-old Krul was brought onto the pitch in the last minute of the extra time. He blocked the second and fifth round shots from Costa Rica's skipper Bryan Ruiz as well as defender Michael Umana while his teammates kept a perfect four-on-four record to seal the hard-fought victory 4-3.
The Netherlands-Argentina match will be held in Sao Paulo Wednesday while the other semi-final match between the hosts Brazil and Germany will be at Belo Horizonte Tuesday.
Before Krul came onto the pitch, it was the Ticos' goalkeeper Keylor Navas who had stolen the spotlight as his seven impressive saves help to maintain a clean sheet for Costa Rica.
The Netherlands started the match aggressively with Robin van Persie, Arjen Robben as well as PSV's promising star Memphis Depay leading the attacking line.
Throughout the first half, the in-form Navas made four saves.
The rhythm of the match was not changed much after the interval. The Dutch side enjoyed the majority of the play trying to break the deadlock while the Ticos were kept busy defending while waiting for chances patiently through counter-attacks and set pieces. And Navas made two more saves.
The Netherlands was awarded with a free-kick from the right flank in the 52nd minute. Robben hit a low pass towards the edge of the box before Sneijder's one-touch effort went over.
In the 82nd minutes, Sneijder curled another fine free-kick from the edge of the area. The shot managed to beat Navas but was denied by the near post. Two minutes later Van Persie broke into the Costa Rica area and snapped a right-footed attempt which was again saved by Navas.
The Netherlands could have finished the match in regular time, but Van Persie's shot from the far post hit Costa Rican midfielder Yeltsin Tejeda, who was standing on the line, and looped up onto the bar before the Ticos cleared it to safety.
And Navas was still on-fire in the extra time. Robben's corner in the 94th minute was received by Ron Vlaar, whose diving header bounced in front of Navas before he palmed it away past the post.
Defending for most of the match, Costa Rica launched an attacking surge in the final five minutes. Second half substitute Marco Urena broke through into the Dutch area but the shot was blocked by Cillessen, which was his only save in the match.
And the Dutch side responded quickly. Sneijder's wonderful right-footed curling effort one minute before the final whistle once again was denied by the woodwork, which was the second time for Sneijder and third time for the unlucky Netherlands.
Higuain's winner came just eight minutes into the game. He received a deflected pass from midfielder Angel Di Maria, turned quickly and fired a flash shot before the Belgian goalkeeper Thibaut Courtois could react. The stunning strike from the top of the area found the side netting at the far post.
Argentina coach Alejandro Sabella said after the game: "After 24 years, we can be among the four best teams in the world. So as you can tell, this is a huge, but moderate joy. We have to reserve our energy."
In the World Cup 2006 and 2010, Argentina had been beaten by Germany twice in the quarter-finals.
Belgium coach Marc Wilmots blamed the Italian referee Nicola Rizzoli after the match, saying, "every time there was a little foul it was always in favour of Argentina. Every time something happened with Messi, the referee gave him a free kick."
