Jorge Sampaoli's side produced an abysmal second-half display that severely damages their hopes of following Croatia out of Group D, with Caballero's stray pass to Gabriel Mercado that was volleyed home by Rebic in the 53rd minute proving the turning point.
Any hopes of a late Argentina fightback were extinguished 10 minutes from time as the irrepressible Luka Modric whipped a fine strike into the bottom-right corner from just outside the penalty area.
Argentina had all but given up by the closing stages and they were punished further in stoppage time when Ivan Rakitic tapped in from close range with barely a defender in sight.
All eyes were on Lionel Messi before the game after his saved penalty in the South Americans' opening 1-1 draw with Iceland and, once again, he was a shadow of the player that has starred for Barcelona.
Messi cut an isolated figure throughout and was unable to rouse his side in the closing stages, leaving them in danger of needing a win and a swing in goal difference to reach the knockout stages.
Simply incredible from #CRO as they seal their place in the knock-out stages with a historic victory!
— FIFA World Cup (@FIFAWorldCup) June 21, 2018
Argentina brought in Marcos Acuna, Enzo Perez and Mercado after the draw with Iceland and started sluggishly, almost falling behind after just five minutes.
Ivan Perisic surged into space down the left and cracked a shot towards the bottom-right corner, which was superbly tipped around the post by Caballero – only starting in Russia due to a knee injury for regular number one Sergio Romero.
Sampaoli's men were far from fluid in attack in the early stages, but did twice come close to breaking the deadlock.
A superb Dejan Lovren block prevented Maximiliano Meza's low strike finding the net and then Acuna's whipped cross clipped the top of Danijel Subasic's crossbar.
Perez squandered a golden opportunity to score at the half-hour mark, conspiring to strike wide of an almost open goal from 15 yards after some haphazard Croatia defending.
Mario Mandkuzic somehow failed to find the back of the net just three minutes later, stealing in unmarked at the back post and flashing a header wide of Caballero's right-hand post from six yards.
There was still time for one more act of profligacy before the interval when Rebic made a mess of Modric's wonderful pass behind the Argentina backline, thumping well over.
There was no such wastefulness from Rebic eight minutes into the second period, as he unerringly rifled a volley past Caballero after the goalkeeper's chip lacked the power to evade him.
Argentina almost bounced back soon after, but Subasic kept out Meza's close-range effort from substitute Gonzalo Higuain's cutback and they were unable to bundle the rebound home.
A leveller scarcely looked likely after that and their miserable night was compounded when Modric curled in superbly from 20 yards.
Rakitic was fouled by Javier Mascherano – Nicolas Otamendi receiving a booking for appearing to kick the ball towards the consequently prone Barca midfielder's head as tempers flared – and crashed the resulting free-kick against the crossbar.
There was no denying Rakitic in the 91st minute, though, as he slotted home from the centre of the box after Mateo Kovacic's initial shot was saved by Caballero, making a premature exit a distinct possibility for the two-time champions.
— OptaJoe (@OptaJoe) June 21, 2018
Photo Getty Images. Caption: Willy Caballero reacts after his mistake against Croatia