The three-time world champions began their Group J campaign with a win, marking the first occasion that Argentina have triumphed in their opening match as reigning champions. Their previous attempts to start a World Cup defence ended in defeats in 1982 and 1990.
Messi, making his 200th appearance for Argentina and his 27th at the World Cup, continued to rewrite the record books. The eight-time Ballon d'Or winner became the first footballer to feature in six World Cups and also moved into a share of first place on the all-time World Cup finals scoring chart.
His tally now stands at 16 goals, drawing him level with former Germany striker Miroslav Klose.
"Speechless. He has been doing this for 20 years; we have to enjoy him for what he transmits to the entire world, it's incredible. We are going to fight match after match," said Argentina coach Lionel Scaloni.
The Argentine star had a goal ruled out before the breakthrough arrived in the 17th minute. After being picked out by Inter Miami teammate Rodrigo De Paul, Messi fired beyond goalkeeper Luca Zidane, the son of French legend Zinedine Zidane.
Algeria also saw an effort disallowed in a lively opening half, while Messi escaped punishment following a reckless challenge on Aissa Mandi.
Argentina doubled their advantage on the hour mark when Messi capitalised on an error from Zidane. He then completed the first World Cup hat-trick of his illustrious career in the 76th minute, striking from the edge of the penalty area after regaining possession from Nicolas Gonzalez.
"Class is permanent, because we're not talking about any old player here, we're talking about a footballer who has won the Ballon d'Or seven or eight times in his career," said Algeria coach Vladimir Petkovic.
"Unfortunately, we also afforded him an opportunity with the first and second goal, and we made it easier for him to score the goals.
"But Messi, with his clear thinking at crucial stages of the game, can do things that much more easily.
"He has the privilege that the entire Argentina team works for him and supports him, and for several years, nay for decades, he's been doing incredible things. Argentina had 10 shots today, and seven of those shots came from Messi."
The defending champions, who are aiming to become only the third nation to retain the trophy, will next face Austria and Jordan in Group J.
