The federal government faces an uphill battle to get controversial elements of its budget, such as welfare changes and a GP co-payment, passed by the upper house.
Mr Abbott received a standing ovation as he took to the stage at the Liberal Party's federal council in Melbourne on Saturday.
"I say to the new senators, we won't hector you and we won't lecture you," he said.
He said he respected the election of the cross-benchers and asked them to respect his.
The new Senate will sit for the first time on July 7 and a Palmer United Party block of senators will share the balance of power with an extended cross-bench.
Australians were fed up with a policy of economic drift, after six years, Mr Abbott said, and he defended the federal budget as purposeful, thoughtful and effective.
He also outlined a game plan to reform the federation by winding back the federal government's role in core state responsibilities.
Mr Abbott also released the terms of reference of a white paper on the matter.
The Commonwealth will continue to take a leadership role on
issues of genuine national and strategic importance but there should be less Commonwealth intervention in areas where states have primary responsibility.
The white paper was promised during the 2013 election campaign as a way to end waste, duplication and buck-passing between Canberra and the states.
It would ensure state and territory governments are "sovereign in their own sphere".
Senior officials from the prime minister's department will work on the white paper in consultation with the states and territories and representatives of local government.
It is due to be released at the end of 2015. A green paper will be completed in the first half of next year.
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