However, special prayers for former Iraqi President,who was hanged in Iraq on Saturday, were held in the capital.
Hundreds of Muslims took part in special prayers organised for Saddam at the Jama Masjid, Fatehpuri and other mosques in the capital.
The celebrations were marred by protest against the hanging of the former Iraqi President by wearing black arm bands by the faithfuls across Madhya Pradesh.
Sunni Samaj district chief Dr Abdul Rashid Patel said after the Id prayers, tributes were also paid to Saddam Hussain by rendering quotes from the Quran.
Earlier, effigies of US President George W Bush were burnt at different places in the state immediately after Saddam was hanged to death on Saturday.
In Mumbai too, the celebrations passed off peacefully with less display of festivities by Muslims in the wake of the sudden execution of former Iraqi President.
Meanwhile, Prime Minister Manmohan Singh called on President A P J Abdul Kalam to greet him on the New Year and Eid-ul-Zuha. Dr Singh, who visited Rashtrapati Bhavan, was closetted with the President for about 15 minutes, an official told UNI.
Jammu and Kashmir Chief Minister Ghulam Nabi Azad prayed for end of the saga of highhandedness, hatred and violence in the state and creation of peace and brotherhood, the essence of Islam for betterment of the people.
Referring to the hanging of former Iraqi President, Mr Azad said he along with his Council of Ministers have described it unfortunate, terming the action as hasty and hurting the sentiments of people.
Meanwhile, Muslims in large numbers offered special Eid prayers in various mosques and shrines amid pleasant weather. The main congregations in the summer capital were held at the historic Hazratbal shrine, housing the holy relic of the Prophet Mohammad, and also at the grand Jamia Mosque.
In Punjab, Muslims thronged various mosques to celebrate the occasion. A large number of them congregated at a mosque at Malerkotla in Sangrur district and offered prayers. Similar celebrations were held in Haryana where the Muslims offered prayers in mosques at different places.
In Chandigarh, hundreds of muslims offered special Eid-ul-Zuha prayers at the Jama Masjid in Sctor 20.
Traditional fervour and enthusiasm marked the occasion in Himachal Pradesh. In Shimla, a large number of Kashmiri labourers offered ''namaz'' at Jama Masjid and in mosques at Boileugunj, Sanjauli and Chotta Shimla. Reports of celebrations were also received from Bilaspur, Chamba, Nahan and Solan in the state.
In West Bengal, the festival was observed with gaiety while thousands of devotees offered prayers on the city's Red Road in the maidan area in a massive assembly.
In Nagaland, the main function was held at the Assam Rifles Ground where over eight thousand followers of Islam congregated from various parts of the state.
People in large numbers offered 'Namaz', which was administered by the 'Pesh Imam' of the Jami Masjid here.
Puducherry Lt Governor Mukut Mithi, Chief Minister N Rangasamy, his cabinet collegues and leaders of several political parties greeted the Muslims on the occasion.
In Tamil Nadu, hundreds of camels were slaughtered for the festival, following Madras High Court vacating its interim stay on camel slaughtering.
Headquarters Secretary of Tamil Nadu Muslim Munnetra Kazhagam (TMMK) Anver Deen told UNI that the TMMK alone arranged 160 camels for slaughtering all over Tamil Nadu. In the city, 32 camels were slaughtered.
Around 1.5 lakh goats, each costing Rs 5,000 and above were sacrificed in Mumbai alone to observe Id-ul-Zuha.
After the prayers, Muslims offer sacrifice of animals. The sacrificed meat is then consumed and distributed among relatives, friends and the poor.
Children, in colourful new dresses, exchange gifts and visit their relatives and receive Eidi (gifts mainly in cash) from their elders.
The Eid-ul-Zuha or Feast of the Sacrifice in Islam, commemorates Prophet Ibrahim's willingness to kill his first born son Ismail for God.
It is celebrated on the tenth day of Zul-Hijja, which is the last month of the lunar Islamic calendar, and also marks the end of the annual Haj pilgrimage.
This year, the festival is being observed two days after its observance in the Islamic world due to lunar calculations and the Indian calendar.