The House last night passed, 282 to 148, a $165 billion plan for shoring up the Social Security System after approving a provision to raise the normal retirement age gradually from 65 to 67 in the next century.

The measure next goes to the Senate, where the Finance Committee will start voting today. Sponsors hope to have a rescue plan at the White House by Easter.

The amendment to save money over the long run by raising the retirement age was offered by Rep. J.J. (Jake) Pickle (D-Tex.), chairman of the House Social Security subcommittee.

It passed, 228 to 202, with strong support from Republicans and over the strenuous opposition of many leading Democrats, including House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) and Rules Committee Chairman Claude Pepper (D-Fla.).

The House bill would postpone this year's scheduled cost-of-living benefit increase from July 1 until next January; raise the Social Security tax rate next January by speeding up increases already scheduled for later years; tax half the Social Security benefits of higher-income recipients for the first time, and bring all new federal workers into the system, along with members of Congress, judges and the president and vice president.

It would increase revenues and cut costs a total of $165 billion over the next seven years, enough to take care of the giant system's short-term problems. It would also eliminate the entire long-term deficit, actuaries say.

In addition, the huge bill would:

* Fundamentally alter the way the government reimburses hospitals under Medicare, by setting up a so-called prospective-payment system under which fees would be set in advance for various services. Hospitals would have to keep their costs below these levels or lose money.

* Extend unemployment benefits, by granting workers who have exhausted their 55-week entitlements under regular or special programs an extra 10 weeks, in some cases, for a total of 65 weeks. Others could get up to 63 weeks.

* Give recipients of supplemental security income--federal aid to the aged, blind and disabled--a $20 increase in benefits July 1 to compensate for cancellation of their July cost-of-living increase. Couples would get a $30 increase.

The Social Security retirement fund will run out of money this summer if Congress does not act. Ways and Means Committee Chairman Dan Rostenkowski (D-Ill.) told the House that the rescue bill was a reaffirmation of "our commitment, this nation's commitment, to its national system of retirement income security."

But Rep. Bill Archer (R-Tex.) said the bill did not go far enough and called hopes that it would solve the system's long-range problems "pie-in-the-sky unreality."

Though federal employe unions lobbied hard against inclusion of future federal employes in the system and demonstrated yesterday in front of a House office building, the federal worker-coverage provision remained in the bill and the House rule governing floor debate did not allow an amendment to take it out.

The bill follows closely the recommendations made in January by a bipartisan presidential advisory commission and endorsed by both President Reagan and O'Neill.

But those recommendations did not fully wipe out the system's long-range deficit, which will arise in part as the population ages and the ratio of retirees to workers increases after the year 2000. Ways and Means moved to close this gap by cutting basic benefits and raising taxes slightly in the next century.

Under the Pickle alternative, the so-called normal retirement age, at which workers are entitled to full benefits, would rise from 65 to 66 in steps from 2000 to 2009, and to 67 from 2017 to 2027. Workers could still retire as young as 62, but with only 70 percent of full benefits; they now get 80 percent. Those retiring at 65 would receive 86.7 percent of their entitlements under current law, and, at 66, 93.9 percent.

Pickle, arguing for his retirement age amendment, said people are living longer and can fairly be required to work longer than in the past.

But Rep. James M. Shannon (D-Mass.) countered that, while workers live longer, they cannot necessarily keep working. He called the Pickle amendment merely a future cut in benefits, since people retiring at 62 or 65 would receive less than under present law because of penalties for early retirement.

After adopting the Pickle amendment, the House rejected, 296 to 132, an alternative from Pepper that would have substituted a further tax increase in the year 2010.

The House bill would:

* Put off this year's cost-of-living increase until next Jan. 1, then provide for such increases each January thereafter.

* Bring under Social Security all federal workers hired beginning next Jan. 1, including new employes of Congress and the courts. There would be immediate coverage of the president, vice president, members of Congress, federal judges and political and executive-level appointees to federal jobs.

* Bring into the system all employes of nonprofit organizations and bar state and local government units from dropping out of the system once they opt in.

* Starting in 1984, make half of Social Security benefits subject to federal income tax, but only to the extent that this half boosts total income over $25,000 for a single person or $32,000 for a married couple filing a joint return. Proceeds of this tax would be fed back into the Social Security trust funds.

* Raise the Social Security tax rate for both employers and employes from 6.7 percent to 7 percent next Jan. 1, on the first $37,800 of income. For 1984, workers would be entitled to income tax credits to offset this increase. By 1990 the rate would rise further to 7.65 percent.

* Starting next year, require self-employed persons to pay the full employer-employe Social Security tax; they pay only part now. This increase also would be offset by an income tax credit.

* If the trust funds dip below stipulated levels after 1987, peg cost-of-living increases to the lower of the previous year's increase in wages or prices across the economy; prices are always the standard now.

* Remove Social Security from the unified federal budget in 1988 and count its revenues and costs separately.

On the final vote, 185 Democrats voted in favor and 79 against, while 97 Republicans voted for and 69 against.

Local members in favor were Maryland Democrats Roy Dyson, Clarence D. Long and Barbara A. Mikulski, Virginia Democrats James R. Olin and Norman Sisisky and Virginia Republicans Herbert H. Bateman, Thomas J. Bliley Jr., J. Kenneth Robinson and G. William Whitehurst.