With a strong showing of support from moderate Republicans, the National Governors' Association today backed President Carter's "far-reaching and audacious" welfare revision and promised to push for early enactment by Congress.

By a 31-to-4 vote, the governors overrode objections from conservatives who said it would prove much more expensive for taxpayers and provide much less relief to the states than advertised.

Republican Mills E. Goodwin Jr., one of the four dissenters, said he was convinced the Carter plan "will increase the number of welfare recipients by a substantial margin, and the states' role and voice will cease and go away completely."

He was joined by Republicans James B. Edwards of South Carolina and Meldrim Thomson Jr. of New Hampshire and Maine's Independent governor James B. Longley.

But the resolution was backed by such moderate Republicans as Michigan's William G. Milliken, Delaware's Pierre S. DuPont IV, Iowa's Robert D. Ray and Indiana's Otis R. Bowen. The administration hopes this will add impetus to the legislation, now awaiting hearings in the House.

Several of the Republicans expressed misgivings about specific provisions. But Ray said the basic principles of the administration plan coincided with the governors' past task force recommendations and said, "We'd be turning backwards if we turned our back on this proposal.

"It doesn't make any difference who takes the credit," said Ray, who was chairman of the 1976 Republican platform committee. "This is basically our program."

While giving the administration a major victory on the welfare issue, the governors rejected the conclusions of a Justice Department study critical of the Law Enforcement Assistance Administration, and they flatly opposed the cargo-preference bill backed by President Carter.

The voting on resolutions came at the end of a day of tours of Detroit police, health and jobs facilities - an effort to confront the state executives with the problems of a big city. The tours were arranged by Milliken, who was elected today as the 1977-78 chairman of the Governors' Association.

In a morning panel discussion, Ray warned that a new Arab oil embargo would be more "catastrophic" than that of 1973, because almost half the states will have not given their governors adequate emergency powers to allocate energy.

David J. Bardin, deputy administrator of the Federal Energy Agency said, however, that the United States is in better shape to withstand severe winter weather than it was last year.

Bardin gave the rundown on fuel reserves.

"Oil supplies are in reasonable shape. The propane supply is down 18 per cent from last year, but the government is encouraging additional imports.

Natural gas storage has passed the 70 per cent mark toward a goal of full capacity by Nov. 1. Coal stock at utilities are close to the 90-day standard the industry looks for."

Bardin promised the governors that the new Department of Energy would have a fuel supply monitoring system ready for next winter and would have standby gasoline rationing plans prepared in case of another embargo.