Aguero himself had opened scoring for the South Americans, before Alfred Finnbogason equalized four minutes later.
Both Portugal and Argentina started their World Cup campaigns with a draw.
Aguero continued: "The first match is always the most hard".
Argentina coach Jorge Sampaoli said: "We need to find more variety of options going forward". "Leo" is yet to win a major global tournament and with his 31st birthday looming, the time is running out for the legend.
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But he couldn't covert from 12 yards with Iceland goalkeeper Hannes Halldorsson denying the Barcelona star.
Then Iceland's defence kicked in, keeping nearly the entire team behind the ball and swarming the Argentina attackers, especially Lionel Messi.
11 MINS: In a run reminiscent of Maradona at his finest, Messi glides and ghosts his way through a sea of Iceland players and sets up a chance for Sergio Aguero that results in a corner. Now they are up against Lionel Messi and it will be fascinating to see how they get on. Argentina suffered a heart-breaking loss in the final to Germany in extra time at World Cup 2014 in Brazil, and they'll be focused on getting back to the final this tournament. "I guess our game plan worked nearly perfectly".
"Maybe we needed to be more creative in the first half". It's a very competitive group.
Messi was visibly frustrated throughout the match, especially in the moments that followed his missed penalty kick. Let us know what you think by leaving a comment below.
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Just as they did two years ago at Euro 2016, Iceland are looking perfectly at home amongst the world's best. Playing under a third different manager of their qualifying campaign, his hat-trick in their last group match against Ecuador was pretty much the only reason they made it to Russian Federation.
Despite playing second fiddle to David de Gea at Old Trafford, Romero has been Argentina's first-choice stopper for the best part of a decade.
It came after some horribly slack play from Argentina and was the type of squandered chance teams so often live to regret. Spinning off a defender with his back to goal at the penalty spot, Aguero moved to his right and hooked his left foot around to send a rising shot high into the net.
Iceland's players rushed to the stands to share their celebrations with their fans and now will have real belief that they can get out of a group that also contains Croatia and Nigeria, who meet later on Saturday.
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The hat trick was his first ever at the World Cup and matched his goal production in his first 13 games over three World Cups. Scoring two goals in the opening, Cheryshev's masterpiece put in the sure-to-be fun scramble for Golden boot.