Koscielny doubled the lead two minutes later from Santi Cazorla's ball in.
Ross Barkley's deflected shot brought Everton back into it, but Romelu Lukaku wasted an excellent, if rare, chance to level as he headed against the bar.
Arsenal goalkeeper Petr Cech denied Gerard Deulofeu late on, but the hosts also had their chances to make the scoreline more comfortable.
Everton's miserable evening was compounded by Gareth Barry's injury-time dismissal as he picked up a second yellow card for a foul on Kieran Gibbs.
Same old Arsenal
Ozil and Giroud were the scorers in Arsenal's Champions League win over Bayern Munich on Tuesday and the two combined brilliantly to break the deadlock after an even opening half hour.But Arsenal's fourth successive win was secured in more conventional fashion than their third.
Giroud and Ozil both hit the woodwork in the second half, with the latter sparkling with invention in a floating role behind the main striker.
Ozil was ably abetted by Santi Cazorla who roamed upfield from his deep-lying position to prompt and probe as Francis Coquelin tidied behind him in midfield.
Problems past, present and future for Everton
It is now 22 meetings and more than 19 years since Everton won away at Arsenal.In more recent history, Roberto Martinez's side were fifth in the table after their win over West Brom at the end of September.
They have picked up one point from three matches since to slide back to 10th.
There are also problems for the future after captain Phil Jagielka limped off with an apparent knee injury after an innocuous challenge with Alex Oxlade-Chamberlain.
The travelling Toffees fans faced a race across London on the final whistle after the last train back to Liverpool - the 21:30 from Euston - was cancelled. The home fans greeted news of the altered timetable with loud cheers.
Man of the match - Mesut Ozil
"Could have gone both ways"
Arsenal manager Arsene Wenger: "At 2-1 we became a bit edgy. We gave a lot on Tuesday night against Bayern Munich and I thought it showed in the last 20 minutes."We couldn't score the third goal and Everton had chances - it could have gone both ways.
"When you win such a big game like on Tuesday it shows a lot about the mentality of your team.
"We have a lot of questions about our strikers but Theo Walcott is outstanding and Olivier Giroud is outstanding at the moment."
Everton manager Roberto Martinez: "I thought we shot ourselves in the foot in giving away such an easy two-goal lead.
"But I very much enjoyed the reaction - the personality and desire of wanting to get back.
"We're very concerned about Phil Jagielka. It's a knee injury - we don't the extent of it, but that's the biggest negative today."
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