Higuain buried a shot into the corner of the net in the eighth minute at the national stadium to set up a last-four clash against 2010 runners-up the Netherlands or surprise package Costa Rica, who meet later on Saturday.
Higuain reacted brilliantly when an Angel Di Maria pass was deflected towards him off Belgium defender Jan Vertonghen, swivelling and striking the ball first time past keeper Thibaut Courtois, as Argentina ended a run of being eliminated in the quarter-finals at the previous two World Cup tournaments.
They last appeared in the semi-finals in 1990, when a team inspired by captain Diego Maradona eliminated hosts Italy on penalties before losing 1-0 to West Germany in the final.
“I knew the goal was going to come and it came at an important time,” man-of-the-match Higuain, who had not scored for his nation in six games, said in a television interview.
“We were able to reach a semi-final which we haven’t achieved for many years,” the 26-year-old added.
“We gave our all, it’s a World Cup, it doesn’t happen every day. Now we need to play well and try and reach the final. There are two matches left to achieve this beautiful dream.”
Belgium had a couple of chances but otherwise offered little in attack despite sustained late pressure that set nerves among the thousands of Argentines thronging the giant arena jangling.
The Europeans’ best World Cup result was in 1986 when they were knocked out by Maradona’s Argentina in the semi-finals.
“We were not impressed by the Argentines, absolutely not, it’s just an ordinary team,” dejected Belgium coach Marc Wilmots said at a news conference.
“There were people in my staff who were crying in the changing, room, we had one or two chances and you have to score at this level,” he added.
“We’ve learned a lot today, we’re incredibly sorry we lost but we are very proud of the boys.
“They shouldn’t regret anything, we were the youngest team among the last eight.”
The light blue and white-clad Argentina fans swamped the 70,000-capacity stadium in the Brazilian capital and were in fine voice while, predictably, there was only a smattering of the red and yellow of Belgium on display.
There was also a sizeable contingent of Brazilians in the crowd who cheered every Belgian touch and roundly jeered their arch rivals.
Argentina looked the more ambitious in the early stages and Higuain’s beautifully executed goal gave them a deserved lead.
They came close to a second in the 28th minute when Messi, who was a constant threat to the Belgium defence having set the tournament alight in the group stages, threaded a superb ball through for the galloping Di Maria.
However, the winger injured himself when his effort was blocked and was replaced by Enzo Perez a few minutes later.
Belgium were starting to enjoy more possession without really looking likely to score before Messi flashed a free kick narrowly wide five minutes before the break.
With Argentina content to sit on their lead, Belgium pushed forward and Kevin De Bruyne let fly from 25 metres midway through the half forcing a save by Sergio Romero before Kevin Mirallas’ diving header went wide from Vertonghen’s centre.
Ten minutes after the break Higuain, who was in fine form, raced through the centre but, with Belgium goalkeeper Thibaut Courtois beaten, his shot clipped the top of the crossbar and the ball flew into the crowd.
Marouane Fellaini sent a 61st-minute header just over the bar in another good chance for Belgium and there was some desperate defending from the Argentines in the closing minutes.
But they held on and captain Messi could even afford to squander a late one-on-one with Courtois as his side progressed along with hosts Brazil and Germany, who went through to the semi-finals on the other side of the draw on Friday.
“I’m very happy for the players who played an excellent game,” Argentina coach Alejandro Sabella told a news conference.
“I’m very happy for them and the Argentines who have waited 24 years for us to be among the best four in the world.”
Argentina’s semi-final is in Sao Paulo on Wednesday, a day after Brazil meet Germany in Belo Horizonte.
(Additional reporting Patricia Avila in Santiago and Patrick Johnston and Brian Homewood in Brasilia, editing by Ken Ferris)