His first half strikes ensured Arne Slot's men secured maximum points against the hosts, who were reduced to 10-men after Jurien Gaari was sent off for a dangerous play.
Idrissi came into this game having scored in Alkmaar's whitewash of Fortuna Sittard last week. He continued his fine form against Fred Grim's team by scoring after 35 minutes thanks to Myron Boadu's assist.
On the stroke of half time, the Moroccan profited from another Boadu again to put his side two goal goals up.
However, he made way for Jordy Clasie in the 61st minute as his team claimed a vital away win.
Comoros' Said Bakari made a 46th minute appearance for RKC after replacing Anas Tahiri, while Dutch-Senegalese Sylla Sow featured for 73 minutes.
Alkmaar remain league leaders with six points and on Thursday, they welcome Mariupol in their Europa League clash before hosting Groningen at the weekend.