Large Eid gatherings were held in Hazratbal Shrine, Eidgah grounds and other places in Srinagar city and also in Baramulla, Kupwara, Ganderbal, Anantnag, Badgam, Pulwama, Kulgam and Shopian districts.
Police said the clashes started after the Eid prayers.
So far, no major untoward incident has been reported from anywhere in the Valley, they said.
Separatist leaders, Syed Ali Shah Geelani and Mirwaiz Umer Farooq were placed under house arrest in Srinagar to prevent their participation in Eid prayers.
Children wearing new clothes accompanying their fathers also came out to attend Eid prayers.
Devotees in large numbers offered Eid prayers in Jammu city as Hindus waited patiently outside the prayer grounds to greet their Muslim brothers.
Large congregation of Muslims were seen offering prayers in Rajouri, Poonch, Kishtwar and Ramban districts of Jammu region.
A highlight of Eid festivities was the communal harmony shown by followers of other religions in the Jammu region who greeted Muslims and also offered sweets to them.