The left-back was withdrawn 20 minutes from full-time in Dublin and is expected to be unavailable until October.
He could miss as many as eight Premier League games and two Europa League fixtures, with the Gunners setting a minimum recovery timeframe of eight weeks.
A short note on their official website regarding his health reads: "Sustained an injury to his left knee. Will be out for 8-10 weeks."
Kolasinac's setback comes just over a week out from the club's season opener at home to Manchester City on August 12.
Having signed for the club on a Bosman deal from Schalke last summer, he made an impressive start to life at the Emirates Stadium, playing regularly in the first half of the season before becoming more peripheral after an ankle injury sustained on New Year's Eve against West Brom.
In total, he has featured 36 times for the Gunners, scoring five goals and creating four more.
Aaron Ramsey is expected to return to training in the week leading up to that match after missing the Chelsea clash with tightness in his right calf.
Arsenal claim victory over Chelsea in midweek thanks to a 6-5 penalty shootout victory, which came after Alexandre Lacazette pinched a last-second goal for them in normal time.
Unai Emery's side complete their pre-season preparations with a friendly against Lazio at the Emirates on Saturday.