Arsenal looked to be heading for defeat after Seamus Coleman and Steven Naismith put Everton 2-0 up before half-time. The Gunners lacked a cutting edge up front as Alexis Sanchez failed to test Everton's centre-back pairing of Phil Jagielka and Sylvain Distin.
But Giroud replaced the Chilean at half-time and went on to score a last-minute equaliser, following a strike by Aaron Ramsey. Afterwards, Wenger thanked his compatriot for guiding the Gunners back into the game.
"We looked much more dangerous (after Giroud came on)," the Arsenal manager said. "I knew we were dominating the game in the second half and we needed some physical presence up front. That presence was very important. He had a very good second half."
Arsenal's French manager Arsene Wenger shouts on the touchline during the English Premier League football match …
WHY ARE ARSENAL WORRIED?
Unfortunately for Wenger, Giroud suffered an ankle injury in the dying minutes and could now miss the second leg of Arsenal's Champions League play-off against Besiktas on Wednesday night.
"It is a big game and I hope we have not lost him," said Wenger of Giroud, who left Goodison Park with a protective boot on his left foot. "In the final minute he stretched his ankle. At the moment it doesn't look too good. "
Sanchez is likely to lead the line against the Turks, who gained a 0-0 draw against the Gunners in Istanbul on Wednesday.
Alexis Sanchez of Arsenal and his team-mates look dejected during the Barclays Premier League match between Everton …
WHAT HAPPENED TO SANCHEZ?
Wenger said he had no option but to withdraw the £35m signing at half-time on Saturday.
"At the moment he is not completely ready physically," Wenger said. "He knows he is not at his best physically but when he is his confidence will come back."
WHAT ELSE DID WENGER SAY?
Wenger's team have gained four points from their opening two games and look a lot more durable than the team that crumbled to Aston Villa on the opening weekend of last season. They also look far more resilient than the side that were hammered by Everton at Goodison Park in April.
"When we got beaten here 3-0 (last season) it could have been many more," Wenger said. "Three was quite flattering on the day. Today we deserved the point we earned. "For us to come back from 2-0 down against a team as good on the counter-attack shows the belief and resilience was there."
Arsenal still look a level below Chelsea and Manchester City, but Wenger insists it is not unreasonable to think his team could win the Premier League this season. "That is our ambition," he said. "We had a big test today and at half-time it was even a bigger test. We responded well. If we can maintain that, of course (we can win it)."
Everton Manager Roberto Martinez looks on prior to the Barclays Premier League match between Everton and Arsenal …
AND HOW DID ROBERTO MARTINEZ VIEW THE RESULT?
The Spaniard hailed his team's performance as "phenomenal". But the Everton boss could not hide his dismay at the way sloppy marking allowed Ramsey to turn home Santi Cazorla's cross, then poor defending permitted Giroud to head in Nacho Monreal's ball into the box.
"For 70-75 minutes it was the perfect performance, very similar to the one we had last time against Arsenal," Martinez said. "The first half was full of dynamic football. Scoring two goals should be enough to control the game in a manner we normally do. Certain individuals, and as a team, dropped our level. The disappointment is the result but the performance was phenomenal.
"There was a drop in our energy levels,. We are not a defensive side. We want to get on the ball but to do that you have to have the energy to do that. The energy levels at the moment are vital for us to perform for 90 minutes."
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