Messi took his total to 18 World Cup goals when he netted in the 38th and 95th minutes having earlier missed from the penalty spot as Argentina secured their second win to move on to six points.
His first strike took him past Germany's Miroslav Klose as the all-time leading scorer in men's World Cup history and his second ensured he moved past Marta's 17 goals in the women's tournament, which had stood as the overall record.
"Beyond anything I'm so happy for the win," Messi said. "It was huge, tough and difficult. It would allow us to be relaxed to what's ahead. All matches in this World Cup are very even, very intense. I'm enjoying this moment and craving to enjoy with my teammates."
"Lionel Messi is the best footballer of all time for me! Congratulations, champion!" Klose told the Süddeutsche Zeitung.
The Inter Miami playmaker, who will celebrate his 39th birthday on Wednesday, showed few signs of slowing down when he claimed his fourth and fifth goals in two matches so far at the finals.
That was despite showing a familiar weakness from 12 yards when he squandered an eighth-minute opportunity to move past Klose's mark, which he had matched with a hat-trick against Algeria.
Lautaro Martinez had been scythed down as he burst into the area by a combination of Stefan Posch and Xaver Schlager although referee Amin Omar needed to view the pitchside monitor before awarding the penalty that Messi dragged wide.
That was the third World Cup penalty miss of Messi's career, the first coming against Iceland in Russia in 2018 and the second versus Poland four years later.
"There were moments when I was really angry about missing the penalty, but I was able to make up for it," said Messi,
David Alaba then twice frustrated Messi, poking the ball towards keeper Alexander Schlager as he shaped to shoot and later blocking a goal-bound strike but there was little the Austrian defence could do to prevent Argentina taking the lead.
Thiago Almada was the architect of the opener when he drove towards the area, laying off possession to Facundo Medina on his left and, after Almada allowed the return pass to roll through his legs, an unmarked Messi calmly stroked home.
Emi Martinez pushed Marcel Sabitzer's set-piece to safety having seen Cristian Romero give away a needless free kick on the edge of the penalty area as Austria sought the equaliser early in the second half.
Michael Gregoritsch headed his later looping effort high over the bar for Ralf Rangnick's side, who rarely managed to muster the quality in the final third to trouble their opponents.
Argentina's second goal did eventually arrive through Messi moments before the final whistle.
Julian Alvarez's initial attempt was saved by Schlager but the rebound was worked to Messi who, after seeing his first shot blocked, pounced to drive in a low strike from six yards out that confirmed the points.
LIONEL MESSI'S WORLD CUP GOALS
1 vs. Serbia-Montenegro, Gelsenkirchen, 2006
1 vs. Bosnia-Herzegovina, Rio de Janeiro, 2014
1 vs. Iran, Belo Horizonte, 2014
2 vs. Nigeria, Porto Alegre, 2014
1 vs. Nigeria, St. Petersburg, 2018
1 vs. Saudi Arabia, Lusail, 2022
1 vs. Mexico, Lusail, 2022
1 vs. Australia, Al-Rayyan, 2022 (R16)
1 vs. Netherlands, Lusail, 2022 (QF)
1 vs. Croatia, Lusail, 2022 (SF)
2 vs. France, Lusail, 2022 (F)
3 vs. Algeria, Kansas City, 2026
2 vs. Austria, Arlington, 2026
With AP
