Argentina suffered a blow after 33 minutes as di Maria went down clutching his thigh. The Real Madrid winger looked in considerable pain as he left the field but was able to take a place alongside his teammates on the bench to watch the remainder of the game.
Messi then went close from a free-kick just before halftime after the ever-unsubtle Marouane Fellaini had made four clear attempts to bring down the illusive striker.
The second half began much like the first had ended, with Argentina carving out the better chances.
Higuain nutmegged Kompany to put himself through on Courtois' goal but just when it looked like he was about to double the Argentine advantage, the crossbar came to the rescue of the despairing goalkeeper.
Again Belgium looked to respond, bringing on Romelu Lukaku for the ineffective Divock Origi -- who in the group stage was the first teenager to score at a World Cup since Messi -- and Dries Mertens for Kevin Mirallas.
With the Argentina defense being organized by the experienced pair of Martin Demichelis and Ezequiel Garay, the Belgians' best chance of success looked like it would come from a set-piece rather than open play.
Fellaini made a nuisance of himself on numerous occasions, pushing, bullying and harassing as defenders at corners and free-kicks.
However an inability to follow the offside rule repeatedly frustrated the Red Devils.
Belgium hadn't scored a goal earlier than the 70th minute in this tournament but as the game progressed, coach Marc Wilmots became ever more desperate on the sidelines.
"We were not impressed by the Argentinians, they were just an ordinary team," said the former Belgium international.
"They can distort the rhythm, they take 30 seconds to take a throw-in and the referee doesn't do anything. If I played that way I would have been destroyed by the Belgium press."
Seeking a lifeline, Wilmots pushed veteran defender Daniel van Buyten -- likely playing his last game before retirement -- forward to operate as an auxiliary striker.
But it was Argentina which came closest to scoring again, though Messi was unable to add to his 2014 tournament tally of four as the onrushing figure of Courtois blocked his dinked strike on the counter attack.
Belgium scrambled forward one final time but when Axel Witzel's ambitious long range effort flew over, Argentina could celebrate.