The Croatia international lasted just 20 minutes of the European fixture before being replaced by Toni Kroos, having partially torn an adductor muscle in his right leg, and left the pitch in tears.
The 23-year-old underwent tests at Sanitas la Moraleja University Hospital on Thursday, with results confirming the extent of the injury.
Real registered a comfortable victory over the Cypriot side, with Cristiano Ronaldo's double followed up by a Sergio Ramos overhead kick on the hour mark.
Kovacic joins a long list of Real players unavailable for their trip to Real Sociedad this weekend, with Ronaldo serving the last game of his domestic ban for pushing referee Ricardo de Burgos Bengoetxea during the Supercopa victory over Barcelona earlier in the season.
Marcelo will also miss Sunday's trip to Anoeta Stadium, having been sent off in the final minutes of Real's 1-1 Liga draw with Levante last time out.
Karim Benzema remains side-lined with a hamstring injury, with Marco Asensio also a doubt.