Goalkeeper Halldorsson dived to his right to turn away Messi's spot-kick as an eventful Group D opener at the 2018 World Cup finished in a 1-1 draw.
Sergio Aguero's left-footed drive put Argentina ahead in the first half, only for debutants Iceland to hit back four minutes later with a goal from Alfred Finnbogason, the country's first at a World Cup.
Messi had the chance to put the South American nation back in front after a foul on Maximiliano Meza, but Halldorsson's guessed correctly to deny the Barcelona forward.
"For me, as a goalkeeper, to play for Iceland at the first game of the World Cup and face the best player in the world with a penalty, it's a big moment," he told the media.
"It's a dream come true to save it, especially as it got us a point that I hope is going to prove important for us to reach our goal and qualify [for the knockout stages].
"I did some homework. This was a situation I knew could come up.
"I looked at a lot of penalties from Messi, and I also looked at my behaviour in the last couple of penalties I've faced.
"I tried to get into the minds and what they'd be thinking about me - I had a good feeling he was going to go that way today."
Heimir Hallgrimsson's side struggled for possession but defended impressively to frustrate one of the leading contenders to lift the World Cup on Russian soil.
"Our gameplan worked really well," Halldorsson added. "We could feel it in the first half that the game was playing out as we wanted it to. They were playing the game in front of us, never having a lot of space.
"Our confidence grew minute by minute, because you never know what is going to happen in a game against Argentina. We felt that as time grew, we had them where we wanted them.
"They maybe grew a bit frustrated as they weren't getting through. We equalised so early too, which was important for the development of the game. I guess our plan worked perfectly."