Not for the first time in the tournament, Alejandro Sabella's strongly fancied team turned in an often disjointed display and a forgettable encounter appeared destined for penalties as extra time neared a close.
But when Lionel Messi surged at a tiring Swiss defence in the 118th minute and produced a perfectly weighted pass, Real Madrid midfielder Di Maria was on hand to put a frustrating display behind him with a clinical low finish.
Blerim Dzemaili came agonisingly close to levelling as Switzerland launched a last-gasp rally but the day belonged to Di Maria and Argentina, who will face Belgium or the United States next in Brasilia on Saturday.
"The heroes are the 23 (squad members) and the coaching staff," the 26-year-old said told reporters. “We gave everything out there.
"We left our heart and soul on the pitch. We were always in the game and we never gave up for a minute.
"I think because of that we deserved to win. I thought we had a lot of chances and we (made) just made one mistake, which forced our goalkeeper into a save.
"We leave as victors because we did well. Thank God for this win, now we have four days to recover for the quarter-final."
Sabella did not necessarily share Di Maria's view that Argentina deserved to go through and credited Switzerland for their showing during the first half.
"Luckily we were able to win the game in extra time," he told reporters at the final whistle.
"The first half Switzerland were on top, then it was us. It was a strange game.
"Now we have to rest up and prepare for the next game."
Midfielder Javier Mascherano underlined the importance of Argentina having players of Messi and Di Maria's calibre in their ranks.
He told the AFA's official Twitter page: "We went, we seek and we had our reward - (a) victory we deserved.
"Fortunately, in the last ball, appeared Messi and Di Maria."
