Koscielny was replaced after only four minutes of the Premier League clash by full-back Kieran Gibbs, just back from a similar ailment.
"He has a groin strain. That's usually three weeks," Arsenal manager Arsene Wenger told reporters at Goodison Park.
Arsenal had taken the lead after only 51 seconds via winger Theo Walcott but the visitors' defensive frailties were again apparent as a poor error by full-back Bacary Sagna allowed Everton to equalise through Marouane Fellaini.
A poor pass by the French international saw Steven Pienaar dispossess Mikel Arteta but the full-back then wasted a second chance to clear as he kicked the ball straight to Fellaini who fired home in the 27th minute.
"We had a good start and after that I felt Everton came back. Their direct game was always a threat with Fellaini," Wenger said.
"We have regrets on the goal we conceded but overall, I believe that 1-1 is a fair result.
"It was a game of remarkable intensity. We needed to show great spirit today to get away with a point because they were up for it. And overall I can only say I'm pleased with the effort we have put in the game."
Sagna began negotiations with Arsenal over a new contract this week as he enters the final 18 months of his deal at the 13-times English champions, who have struggled to retain their leading lights in recent seasons.
Robin van Persie, Samir Nasri and Mathieu Flamini have all left in recent years after failing to extend deals with the club and Arsenal fans fear Sagna and 23-year-old Walcott, whose contract expires at the end of the season, could be next.
Walcott has struggled for consistency in his seven previous years at Arsenal but his goal at Everton was his 10th in all competitions this season to continue an impressive start to the campaign.
Arsenal have struggled for goals at times this season after selling Van Persie to Manchester United and Walcott, despite not starting in his preferred role as a central striker, is the club's leading marksman.
"We [are building] a new team. Hopefully we can go from strength to strength. We have the spirit. I feel we have some room for improvement in the final third," Wenger said.
"Overall, when you have the spirit, you have a chance. We have to show that now. We have to be consistent," the manager said with an eye on Saturday's match at home to Swansea City.
It was a third draw in four Premier League games for Arsenal and a sixth in eight for Everton with the home side's manager David Moyes feeling his side did enough to take all three points on Wednesday.
Moyes felt Everton should have been awarded a second half penalty when his former player Arteta pulled back Pienaar in the box.
"We should have had a penalty. I thought we played against a really good Arsenal team and a lot of games are won by decisions referees make or don't make," the Scot told reporters.
"We got nothing tonight whatsoever and the players had to earn everything they were going to get because we weren't going to get a lot, that's for sure."
