Sitting at the bottom of Group D after failing to pick up a point in their first two matches, Australia had to win on Sunday morning (AEDT) to remain a chance of making the final 16.
The Young Socceroos did just that, with Melbourne City star Caputo drawing first blood in the 21st minute.
Daniel Bennie added another in the 39th minute, before Caputo popped up again in the 50th to complete his double at Estadio Nacional Julio Martinez Pradanos.
Cuba fought back through a 63rd-minute goal from Alessio Raballo, but couldn't pull off the comeback they needed to prevent their exit from the tournament.
Australia's victory on Sunday is their first win of the event, with Trevor Morgan's side moving to third (1-2) in Group D.
The Young Socceroos haven't progressed past the group stages since the 2003 tournament and face a nervous wait ahead of Monday's final group matches.
Argentina lead Group D ahead of second-placed Italy, with the two countries playing in another match on Sunday.
The Young Socceroos burst from the blocks against Cuba, having five shots on target to two in the opening half.
Australia finished the match with 13 shots on target to Cuba's five.
James Overy won possession in Australia's half and set up Caputo for the opening goal with a brilliant through ball.
Caputo, on return to the starting line-up, timed his run brilliantly to beat the outstretched foot of goalkeeper Yurdy Hodelin.
The Young Socceroos found their second goal when Bennie capitalised on a defensive lapse from Cuba.
Cuban midfielder Diego Catasus intercepted a ball from Louis Agosti but struggled to react under hot pressure, allowing Bennie to pounce.
Raballo denied the Young Socceroos a clean sheet after edging out goalkeeper Steven Hall to put the finishing touches on teammate Samuel Rodriguez's cross.
Cuba earlier had tried to find a lifeline when midfielder Maikol Vega's shot hit Brisbane Roar centre-back Lucas Herrington's arm in the 55th minute.Â
But it was ruled Herrington's arm was in a natural position.
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