The House yesterday overwhelmingly approved a bill to cut Social Security disability benefits an average of 14 percent, but only for families going on the rolls after July 1.

The final compromise version, backed by the Carter administration on grounds it will save an estimated $1.1 billion a year in benefit payments by 1985, now goes to the Senate, where easy approval is also expected.

For years, disability has been one of the fastest-growing social programs in the government, though recently, it has leveled off somewhat. In 1970 there were only 2.7 million workers or their dependents on the rolls and the program cost $3 billion; today it has 4.9 million and the fiscal 1981 cost is estimated at $17.4 billion. At some time in their lives, three out of every 10 workers entering the labor force will draw disability benefits of some kind, experts estimate.

The bill was designed to slow the explosive growth in outlays and to avoid having benefits so high that in the opinion of some, they served to discourage workers from going off the rolls and resuming work.

Former health, education and welfare secretary Wilbur J. Cohen, chairman of a coalition of organization of the disabled and aged that opposed the bill, recently urged President Carter to reverse himself and veto the bill to avoid becoming the "first Democratic president" to approve reductions in Social Security payouts. But Carter is expected to sign the bill.

While cutting benefits, the bill also would make important changes in Medicare and other regulations affecting disabled workers -- all designed to encourage them to return to work. In addition, it would authorize a new voluntary program under which the government would certify commercial "Medigap" insurance programs as meeting minimum standards, and bar Supplemental Security Income (welfare) payments to aliens for the first three years after entry unless the allien's immigration sponsor lacked the money to support him.

By far the most important provisions in the bill are in disability insurance benefit cuts. These benefits are paid workers under 65 who are so disabled they are unable to work at virtually any job.

The benefit formula can be generous, particularly for workers with dependents. Former HEW secretary Joseph A. Califano Jr. used to point out that, in some cases, a disabled worker plus his dependents received combined benefits higher than he previously took home in pay from his job.

The bill approved yesterday would bar a disabled worker from obtaining combined monthly benefits for himself and dependents in excess of 85 percent of his previous average monthly earnings, or in excess of 1 1/2 times his won basic benefits, whichever is lower. These lower benefits would apply only to those going on the rolls after July 1. Those already on the rolls would continue to receive benefits at higher levels.

In addition, the bill would reduce the number of low-earning "dropout years" that may be ignored in computing the disabled worker's basic benefit. At present, everyone may drop out five years, but the bill provides that workers under 47 may drop out fewer years, thus reducing their benefit levels if they become disabled before 47.

For some families going on the rolls after July 1, the new provisions would mean benefits up to one-third lower than under the present formula. But for most, the cut would be about 14 percent. For example, a worker with average earnings of $7,500 a year and two dependents would get $626.20 a month under present formulas, but under the new formula the figure would be $536.

Similarly, a $14,250 annual wage would mean benefits of $979.60 a month under the existing formula but $839.70 under the new formula.

According to the Congressional Budget Office, the government will save a net of only $6 million from passage of the bill in fiscal 1980, but this will rise in steps to $1.12 billion a year by 1985.