With the Croatian defense withdrawing, Álvarez fended off weak challenges from Josip Juranović and Borna Sosa and then slotted the ball past Livacović, sending the huge Argentina crowd into delirium.
Alexis McAllister then went in with an angled header from a corner, forcing a fine save from Livakovic to avoid the third.
Before the break, all Croatia could muster in response was a low Juranovic cross-shot which Emiliano Martinez responded well.
Croatia coach Zlatko Dalić responded with a double substitution at the break, introducing Nikola Vlasic and Mislav Orsic and five minutes later throwing on striker Bruno Petkovic.