Argentina stopper Willy Caballero produced a goalkeeping howler in the second half to leave Ante Rebic with the chance to volley his side into the lead, before Luka Modric went one better with a standout finish from range to leave Croatia in the driving seat.
A memorable night for Zlatko Dalic's charges was capped off during injury time, when Ivan Rakitic poked home a rebound to cement all three points and send Vatreni into the second round with a game to spare.
Defeat for Jorge Sampaoli's men now sees Argentina facing the possibility of exiting the tournament without getting out of the group stages should results go against them.
A tactical switch to a 3-4-3 almost saw Argentina come undone early on when Ivan Perisic broke free inside the box, only for Caballero to pull off a fine save to tip the ball away from his far post.
Argentina faltered when moving forward in the opening half and there appeared little improvement in the final third from La Albiceleste following their lacklustre 1-1 draw against Iceland.
Lionel Messi was rarely in the picture, touching the ball just 20 times in the first half, but he almost carved out an opening when he found space to feed a pass across the box for Maximiliano Meza, whose shot was blocked by a last-ditch challenge from Dejan Lovren.
Argentina's logistical changes did allow Marcos Acuna room down the left flank and the wing-back came close to opening the scoring when his cross drifted awkwardly before clipping the crossbar, as Danijel Subasic was left stranded off his line.
It took the best part of 30 minutes for Argentina to have a clear-cut chance and they should have gone ahead when Croatia failed to clear their lines, leaving Enzo Perez to curl wide from the edge of the box with goalkeeper Subasic caught out of position.
Croatia responded straight away with a crucial chance that should have been buried by an unmarked Mario Mandzukic, although his diving header went wide of the post after he had beaten the offside trap to meet a cross from right-back Sime Vrsaljko.
Both sides were guilty of defensive lapses in concentration and Croatia might have made Argentina pay just before the interval, but once again a lack of quality intervened when Rebic fluffed his lines having been set away on the counter by an inventive Modric.
In a game littered with errors, Argentina were finally made to pay in the 53rd minute when stopper Caballero produced a howler to present Rebic with a difficult chance inside the box.
The keeper sought to chip the ball over the forward's head and find Mercado in space, only for youngster Rebic to catch the loose ball mid-flight and send a spectacular first-time volley back over the goalkeeper to make it 1-0 to Dalic's men.
Substitute Gonzalo Higuain at least added some industry to Argentina's forward line and the striker went on an impressive run to cut the ball back for Meza, who saw his close-range effort somehow kept out down low by Subasic.
That effort proved to be the last of any real note from Argentina as the game wore on and they were eventually punished on a woeful evening, as Modric wound up from range to bend a memorable effort into the bottom corner to make it 2-0 in the 80th minute.
The torment was not over just yet for the travelling Argentina support, Rakitic returning late during stoppage time to prod home into an empty net and register a third goal, after Caballero had produced a strong save to keep out his initial shot.
ARGENTINA (3-4-3): Caballero; Mercado, Otamendi, Tagliafico; Acuna, Mascherano, Perez (Dybala 68'), Salvio (Pavon 56'); Aguero (Higuain 54'), Messi, Meza
CROATIA (4-2-3-1): Subasic; Vrsaljko, Strinic, Lovren, Vida; Rakitic, Brozovic; Modric, Perisic (Kovacic 82'), Rebic (Kramaric 57'); Mandzukic (Corluka 90')