Higuain, now the tournament's top marksman, scored his goals in the 4-1 win over South Korea in Group B at Soccer City that put Diego Maradona's team on the verge of the second round.
"I'm happy to have scored. You have to keep a cool head," said Higuain, who missed two good chances on his World Cup debut when Argentina beat Nigeria 1-0 on Saturday.
Batistuta is the only player to have scored hat-tricks in successive World Cups - against Greece in the United States in 1994 and during a 5-0 rout of Jamaica in France in 1998.
Guillermo Stabile hit the first World Cup hat-trick for Argentina on his debut in a 6-3 rout of Mexico at the first finals in Uruguay in 1930. Stabile, who died in 1966, has the remarkable record of eight goals in four internationals.
"The important thing is for the team to keep winning and, in this fashion, finishing chances which we could not do last time (against Nigeria)," Higuain told reporters after Argentina's impressive attacking performance on Thursday.
"We showed we're in form for what we came here (to do), to win the World Cup. That's the most important thing for us... To achieve that we have to keep improving."
Higuain, the second-top scorer in Spain behind Argentina team mate Lionel Messi with 27 in La Liga last season, has Batistuta's record to aim for as the country's all-time top scorer with 55 in 75 matches. He scored 10 at three World Cups.
The Brest-born 22-year-old, who has five goals from seven international matches, might have opted to play for France.
He chose Argentina, the land of his parents, where he spent all of his life apart from the first few months and before his 2007 move to Real Madrid from River Plate.
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