The two unions account for the major chunk of the autorickshaw and taxi drivers in the national capital.
Rajendra Soni, general secretary of both the unions, said Delhi Transport Minister Gopal Rai has given them assurance.
The strike was proposed in protest against app-based cab operators, and eight other unions which met the minister Sunday had already "distanced" themselves from the stir.
"Today, the transport minister called us for a meeting and he gave us written assurance that the government will take strict action against illegal operations of app-based taxi operators," Soni said, adding, the government has also given written assurance on issuing 1,000 new permits for autorickshaws.
Earlier in the day, Rai had urged Lt Governor Najeeb Jung to direct police to prevent any untoward incident during the proposed strike.
Rai also met senior police and transport department officials for discussing necessary arrangements to tackle any problems arising out of the strike on an even day (April 18), when only even-numbered vehicles, barring those under exempted categories, will be on the roads.
A section of auto and taxi unions including Delhi Autorickshaw Sangh and Delhi Pradesh Taxi Unions had given a call for it, a move which may have caused hardship to commuters in the national capital, where the second phase of odd-even scheme is in operation.
Rai also met senior Delhi Police officials and asked them to deploy more men at places like railway stations and hospitals so that auto and taxi drivers could work freely, he said.