Barkley, the most impressive player in a pulsating Cup tie, had set up Everton's equaliser with a 60-yard run and pass before being presented with a clear shooting chance early in the second half with the score at 1-1. The young midfielder blazed narrowly over from 18 yards.
"I have no complaints," said Martinez, who won the FA Cup with Wigan last season. "Arsenal played really well today, they took their chances and they were very clinical.
"I just felt the key moment of the game would be the second goal and we started the second half really well and we made it a very uncomfortable game for Arsenal.
"We had a really good chance and Ross Barkley in those situations - nine out of 10 times he hits the target. You have to be clinical, and Ross is one of the best players in this league to take advantage of those moments."
Ozil ended a troubled week, and a period of lacklustre form for Arsenal, by opening the scoring after seven minutes.
One-touch football opened up Everton and after a defender slipped, Santi Cazorla was left free to set up the German forward, bought from Real Madrid in August, on the left of the area and he eased the ball past Joel Robles.
Ozil, who was booed off by Germany fans on Wednesday after setting up the winner over Chile in a friendly, looked like a world-class player again, playing with his head up and dragging defenders every which way.
"Of course the goal was important for him and how he took his chance - I hope that will encourage him. What I like about him was that he looked regenerated, he had more power in his runs and he did a lot of dirty work, like tracking back," Wenger said.
Everton were on the ropes after Ozil's goal and Yaya Sanogo and Thomas Vermaelen had shots pushed away by the visiting goalkeeper.
But Barkley's ability to turn desperate defence into flowing attack was evident from early on as loose passing by Arsenal when fully committed around the Everton area set up a clutch of dangerous counter-attacks by the men in blue.
Robles turned a cross by Bacary Sagna on to the bar halfway through the first half, before Barkley's lung-bursting run out of his own half was followed by an excellent diagonal pass into the area. Kevin Mirallas seemed certain to score but he sliced the ball, before Romelu Lukaku stabbed home from a couple of yards out.
Lukaku's strength and quick feet constantly troubled the Arsenal defence and a glorious pass by the Belgian set up his compatriot Mirallas, but he shot tamely at Arsenal goalkeeper Lukasz Fabianski.
The excitement and attacking football was maintained during the second half but Arsenal had learned their lesson and did not commit as fully when on the offensive.
Vermaelen, the club captain but second choice to Laurent Koscielny in the centre of defence this season, put the home side in trouble when he slipped on the touchline when deciding how to clear the ball. Lukaku took the ball to the edge of the area before setting up Barkley for his decisive miss.
The introduction of France striker Giroud for Sanogo gave Arsenal a second wind and, after Cazorla's shot was saved at full stretch by Robles, Alex Oxlade-Chamberlain broke towards the right goalline.
The England forward shimmied around Gareth Barry but the Everton player left his leg in to bring the England midfielder down and concede a penalty.
Arteta hit the back of the net from the spot but the kick had to be retaken, amid huge protests from Arsenal, because Giroud had encroached in the area. But the Spaniard again made no mistake the second time to put Arsenal ahead.
A fine Arsenal move settled the game. Mathieu Flamini worked the ball out of defence, substitute Tomas Rosicky sent right back Sagna to the line and his cutback gave Giroud a simple tap-in.
With Everton committed to attack, Cazorla put Rosicky away down the right and he slid a pass to Ozil in the middle. The German flicked the ball sideways for Giroud to slide home his second.
"It was a quality performance from the first minute to the last, we were unlucky to be 1-1 at halftime," said Wenger. "In the second half Everton started well, they had a strong moment in the game where they had a chance with Barkley, and then we scored the second goal."
- Sports & Recreation
- Olivier Giroud
- Ross Barkley