Coach Lionel Scaloni's side set a new national record of 32 matches unbeaten with the 3-0 Finalissima win over Italy on Thursday (AEST), and it never looked like failing to extend that run – the longest currently intact in international football – against a limited Estonia outfit.
Messi got Argentina on its way from the penalty spot, before doubling his tally with a sumptuous finish on the stroke of half-time.
He completed a treble early in the second half, before adding further goals inside the final 20 minutes as Argentina cruised to victory at Estadio El Sadar in Pamplona, Spain.
Argentina wasted little time in stamping its authority on the game, Messi stroking home from the spot in the eighth minute after Estonia goalkeeper Matvei Igonen wiped out German Pezzella.
Messi grabbed his second in the 45th minute, the Paris Saint-Germain forward superbly curling over Igonen from 10 yards after being played in by Alejandro Gomez.
Argentina's captain and talisman sealed a hat-trick two minutes after the interval, steering home Nahuel Molina's cross from the right wing from 10 yards.
Julian Alvarez and Rodrigo de Paul went close for La Albiceleste, before Messi netted a fourth with a cool finish past Igonen in the 71st minute after taking advantage of uncertainty in the Estonia defence.
He put the seal on a remarkable individual performance to make it 5-0 five minutes later, slotting home from 12 yards after a scramble in the opposition penalty area.