Authorities say nearly 1.2 million registered vehicles in Delhi will be off-road every day during the two-week restrictions.
Air pollution peaks at this time of year as farmers set fires to clear land and traditional Diwali celebrations end with people setting off fireworks.
The 20 million residents of New Delhi, already one of the world's most polluted cities, have been suffering for weeks under a toxic haze that is up to 10 times worse than the upper limits of what is considered healthy.
An Indian traffic police officer holds his hand over his mouth to be extra safe as he patrols along a quiet street in New Delhi on November 2.
"I feel like moving out as well because I'm young and I'm still on a stage of building up my life and my career", resident Divyam Mathur said.
Capt Amarinder had instead blamed Delhi Chief Minister Arvind Kejriwal and accused the latter of playing politics.
But national environment minister Prakash Javadekar accused Kejriwal of politicising the issue, while an MP from Prime Minister Narendra Modi's ruling Bharatiya Janata Party (BJP) violated the odd-even auto rule as a "symbolic protest" by driving a vehicle that was barred under the scheme.
In a ruling following petitions filed by activists, the court's judges ordered an immediate halt to the practice of farmers burning crop stubble in the states surrounding the capital.
The apex court, which also summoned the chief secretaries of Punjab, Haryana and Uttar Pradesh on November 6, said every year a "piquant situation" of severe pollution arises in the region for substantial period and the Delhi government, civic bodies and other state governments concerned are not performing their duties as enjoined upon them.
The vehicle restrictions have been the city government's pet project to fight air pollution.
"The government must roll out a policy so that the vehicles can be reduced on roads".
"This is a shocking state of affairs in which we are put, as on today", the court said. The court also banned construction or demolition activities in the capital until further orders.
Anything above 400 on the AQI poses a risk for people with respiratory illness and can also affect even those with healthy lungs.