The Gunners set up a semi-final meeting against either Reading or Bradford thanks to a 2-1 win over 10-man United at Old Trafford.
Former United striker Danny Welbeck returned to haunt Van Gaal by scoring the winner after Wayne Rooney had equalised Nacho Monreal’s opener.
The main talking point from the match was the latest episode in what has been a largely disappointing start to Di Maria’s career at Old Trafford.
After being cautioned for simulation, the former Real Madrid midfielder remonstrated with referee Oliver and then grabbed the back of his shirt and yelled at the official when he had turned his back on him.
Replays showed the first booking on Di Maria was harsh and Aaron Ramsey had pulled the player back, but the ??59.7million man’s actions thereafter deserved sanction, according to Van Gaal.
“I think he’s touched the referee and that’s forbidden in every country, so he has no excuses,” the United manager said.
“In Spain he knows that he doesn’t touch the referee, but that is also in his emotion.
“I’ve already spoken with him, he knows my opinion but also I have to see on the video.”
Di Maria actually started the game reasonably well, setting Rooney up for his goal with a wonderful curling cross.
But his dismissal clearly angered his manager and the player himself, who refused requests for an interview post-match.
Arsene Wenger was delighted as Welbeck struck a second-half winner.
The Gunners took the lead midway through the first half as Alex Oxlade-Chamberlain weaved his way through the United defence before teeing up Nacho Monreal to coolly slot his close-range shot past a diving David de Gea and into the bottom corner.
It did not take long for the hosts to level matters and Wayne Rooney did the damage as he directed Angel di Maria’s beautiful cross into the back of the net via a well-timed header.
But Welbeck handed Arsenal the lead for the second time as he beat De Gea to the ball just outside the 18-yard box and tapped into an empty net just after the hour mark.
Wenger admitted he was proud of how his side executed their game plan, and told BBC 1: “(We wanted) to start without apprehension but we were caught a few times here.
“(We tried) to be on the front foot from the start and close them down, play with a high pace and I felt that we did that well.”
Arsenal were drawn against Bradford or Reading in the last four and Wenger added: “Last year we played a tricky game at Wembley in the semi-final and you expect that again.
“We have played Bradford before, we know about that but let’s enjoy tonight.
“I feel we played a good game tonight from the start and we deserved to win the game.”
Wenger was full of praise for Welbeck, who left his boyhood club for north London last summer, and provided the killer touch tonight after a blunder from Antonio Valencia.
The manager said: “I believe he is just happy to score. He worked very hard today and he was rewarded by a good goal.”
The Gunners went to Old Trafford on the back of a fine run in the Barclays Premier League, with seven wins in the last eight games putting them third, now a point ahead of United and only four behind Manchester City.
But they are battling to stay in the Champions League after a 3-1 first-leg loss at home to Monaco two weeks ago which they need to turn around a week on Tuesday.
Wenger continued: “We live in the real world and we play game by game, we want to compete in every competition and I believe the result today can give us some good morale.”
Welbeck appealed for a penalty as he went to ground after a touch from Marcos Rojo just inside the 18-yard box during the first half, but Oliver waved play on. Wenger said he thought it was a spot-kick, but admitted he was pleased with his side’s performance despite not winning that appeal.
The Frenchman added: “We had (a shout for a penalty). I think what is the most important when you’re a manager is that the team performs at a good level – and that’s what we did.”