The visitors took the lead as early as the third minute when Mikael Lustig steered Gonzalo Higuain's shot into his own net and the Swedes found it hard to get a grip on the game as their opponents retained possession with ease.
Sweden equalised when Jonas Olsson headed home Kim Kallstrom's corner after 17 minutes, but Sergio Aguero restored Argentina's lead almost immediately with a deft chip.
The South Americans went 3-1 up in the 23rd minute when Higuain was on hand to fire home after Andreas Isaksson could only parry Messi's powerful shot.
Rasmus Elm claimed a late consolation in added time for the Swedes as they fell to their first defeat at the Friends Arena, firing a thundering free-kick into the top corner with what was almost the last kick of the game.
"We scored the goals quickly and that gave us calm to get good control of the ball. The important thing was to try to play well and win," Aguero told reporters.
"We had a good match. Pity about the last goal which embelishes a result that should have had a bigger winning margin for us," said midfielder Javier Mascherano.
"It was as good as I'd expected, a skillful side, many skillful individuals," Sweden coach Erik Hamren told a news conference.
"You can choose to park the bus and chance on the counter, but we decided to try to play. The problem when you meet such individually skillful players is that if you lose the ball as we did for the first goal, you get punished.
"We were a little unlucky that it hit Lustig's foot when we'd backed up the way we should have," said Hamren, who professed himself happy that his side had "won" the second half against tough opposition for the third game in a row.
MESSI DANGER
Despite not getting on the scoresheet and seeming at times to stroll around the pitch, Messi was a danger throughout, fizzing into life when the ball reached his foot.
His opposite number, Sweden captain Zlatan Ibrahimovic, had a quieter night, often drifting out to the left to seek space to work in before being substituted at half-time.
Ibrahimovic's replacement Mathias Ranegie made little difference to the Swedes and there was no shortage of chances for Argentina in the second half.
Messi thought he had got on the scoresheet when he lifted the ball over Isaksson, but the Swedish goalkeeper recovered to make an acrobatic clearance off the line.
Moments later Aguero could have scored a fourth but he chipped just wide after rounding Isaksson.
With Sweden's main threat coming from Olsson's aerial ability at set pieces, Argentina dominated possession for long spells with their quick passing, leaving their hosts chasing shadows.
Sweden face Ireland at home when the World Cup qualifiers resume on March 22, with Argentina hosting Venezuela in Buenos Aires on the same date.
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