Sergio Aguero opened the scoring with a rasping first-half finish to register his first ever World Cup goal, before Iceland hit back immediately to equalise through frontman Alfred Finnbogason just two minutes later in the Russian capital.
In a game of vastly opposing styles but very slim margins, Lionel Messi had the chance to secure Argentina three points from the spot in the second half, although Icelandic stopper Hannes Thor Halldorsson produced a remarkable save to earn his side an unforgettable point.
Argentina started the contest with plenty of energy and they might have found an early lead when Messi turned a free kick into the middle of the box, leaving Nicolas Otamendi to leap highest and send a header wide of the far post.
Iceland aimed to play on the counter for much of the first half, and frontman Finnbogason planted a seed of doubt when he rolled a tame Marcos Rojo on the edge of the box, before volleying over the crossbar.
It was the debutants who arguably had the first major chance when Argentina failed to clear their lines, allowing a ball to be threaded into the path of an onrushing Birkir Bjarnason, who could only drag his shot wide of Willy Caballero's near post.
Iceland defended valiantly in the first half and were able to force a constantly-probing Messi to drop deep, but the first-timers could not stop the quality of Manchester City striker Aguero, who was soon to light up proceedings.
The veteran striker produced a memorable piece of quality in the 19th minute to bring Rojo's wild effort under control with his back to goal, allowing the frontman to then turn and smash home his first ever World Cup goal to make it 1-0.
In typical Icelandic style, Heimir Hallgrimsson's charges rallied straight from kickoff and they were soon to restore parity just two minutes later.
Gylfi Sigurdsson crossed into the middle from the right, where Caballero could only parry his save into the middle of the box, leaving Finnbogason to fire home into an empty net.
Iceland's dogmatic approach almost saw them into a lead just before the break, this time as some energetic pressing allowed Sigurdsson to force his way past two defenders, only to see his cute shot tipped away by a diving Caballero.
The early stages of the second period mimicked the first half, with Argentina holding much of the ball while Iceland sat back and sought to hit La Albiceleste on the break.
As Argentine pressure began to tell on the Icelandic defence, a long ball forward caused problems for Hallgrimsson's men as Hordur Magnusson conceded a penalty for a beleaguered shove on Maximiliano Meza in the 64th minute.
The decision allowed Messi to step up and potentially push his side back into the lead, although Icelandic stopper Halldorsson had other ideas as he guessed correctly by diving to his right and getting two fists on the effort to keep the scores level.
The game was still to be plagued by more controversy when substitute Cristian Pavon went down under a clumsy challenge from Birkir Saevarsson in the latter stages, only for the referee to wave play on and get proceedings moving again, preventing a VAR decision from being made when a penalty might just have been given after a second look.
Argentina could have snatched three points in the final embers of the game when Pavon came in off the left flank to bend an effort through a number of bodies, leaving Halldorsson to again make another stunning save to push the ball away from the far corner and earn Iceland a famous point in their debut World Cup contest.
ARGENTINA (4-2-3-1): Caballero; Salvio, Otamendi, Rojo, Tagliafico; Biglia (Banega 54'), Mascherano; Meza (Higuain 84'), Messi, Di Maria (Pavon 75'); Aguero
ICELAND (4-4-1-1): Halldorsson; Saevarsson, Arnason, Sigurdsson, Magnusson; Gudmundsson (Gislason 63'), Gunnarsson (Skulason 76'), Hallfredsson, Bjarnason; Sigurdsson; Finnbogason (Sigurdarson 89')