It looked as though Messi's moment had arrived on 78 minutes as he picked up the ball on the edge of the box and jinked onto his wand of a left foot.
But just as the thousands of Argentina fans inside the stadium prepared to hail their saviour once more, Benaglio flung out his right hand to make a superb save and ensure the game remained goalless.
As Algeria had done the previous evening, Switzerland burrowed away with the prospect of an extra half an hour the prize at the end of a long, weary tunnel.
And despite a few nervous moments, when the right weight of pass or split second decision could have fashioned a late winner, the Swiss saw it through to the end of 90 minutes.
The extra half hour continued in the same vain, the Swiss content to soak up pressure in a congested final third and look for counter attacking opportunities.
Benaglio was called upon to save from a glancing header from Rodrigo Palacio while Messi was lucky to escape a booking after a tangle with Valon Behrami.
Di Maria, who enjoyed such a stellar end of season with European Champions League winners Real Madrid, cut a profligate figure through the game, but tested Benaglio with a stinging drive from range.
And his big moment would arrive, as so many of Argentina's have, via the boots of Messi.
The three-time FIFA Ballon d'Or winner drove towards a flagging Swiss defense before offloading to Di Maria, whose precise finish back across Benaglio found the far corner.
That sparked delirium among the hordes of traveling Argentina fans but its side's leaky rearguard almost offered up another twist in the game's dying stages.
With Benaglio installed as center forward, a succession of crosses flowed into Argentina's penalty area.
From one, midfielder Blerim Dzemaili found himself unmarked and five yards from goal when the ball arrived at his eye level.
But his header cannoned off the post with Romero stranded and struck him once more, only for the ball to trickle wide of the upright.
Shaqiri then won a free kick right on the edge of the penalty area as nerves at both ends shredded yet further, but once the wall had repelled his effort the final whistle sounded.
Coach Alejandro Sabella stuck to the classic football adage of taking one game at a time. "Our dream is only to work for the next match and try to move onto the semifinals," he said.
"We do not look beyond that. To do so would be a mistake as we have already seen good teams like Spain, Uruguay, Italy and England have gone out."
Argentina limps on, but winning ugly won't matter a jot to its players or fans should a first World Cup in 24 years result from it.