The Senate is heading for its first test of strength Tuesday on the resolution giving the states three more years to complete ratification of the Equal Rights Amendment.

Thirty-five of the required 38 states needed to make the ERA part of the Constitution have approved it. But the seven-year period for state ratification, which Congress allowed when it first passed the constitutional amendment, will expire in March.

ERA backers want more time, and a resolution to give the states an additional three years and three months has passed the House. However, Sen. Jake Garn (R-Utah) and others have threatened to filibuster the proposal in the Senate.

Yesterday, Majority Leader Robert C. Byrd. (D.W. Va.) moved in a lightning stroke to see how many votes each side has - and whether backers of the ERA have the 60 Senate votes needed to end a filibuster.

The vote won't be a final one on ERA, but it will be a procedural test of strength and a crucial step toward final approval.

With only a handful of senators on the floor during a lull in debate on the Labor Department money bill. Byrd suddenly moved to consider the ERA resolution and immediately slapped a debate-limiting cloture motion on the question of whether the ERA measure should come before the Senate.

The Senate will vote Tuesday on whether to limit debate on the motion to consider. If the required 60 votes are obtained. then the ERA-extension measure will, in effect, be the business before the Senate.

Opponents then could mount a filibuster against passage, but Sen. Birch Bayh (D-Ind). Senate sponsor of the ERA measure, said a success in limiting debate on whether to bring up the ERA proposal would be a good indicator that the second filibuster could be cut off, too.

"We'll vote(Tuesday) on whether to take it up," Byrd said. "I would assume the backers of ERA ought to be able to produce 60 votes for cloture. They say they have them."

Bayh said he didn't know exactly where the votes lie on one crucial question: Whether a state that already has ratified ERA can rescind its ratification. Most ERA backers oppose any attempt to add language allowing such action.

Senate Republican Leader Howard H. Baker (R-Tenn.), who strode into the chamber shortly after Byrd laid down the cloture petition, expressed anger over Byrd's move and said it had "chilled our efforts" to rerach agreements with Republicans for voluntary time-limits on amendments to the Labor-HEW bill.

Byrd had told a reporter earlier that failure to speed up action on Labor-HEW amendments appeared to be in part a slowup disigned to delay matters generally and give ERA opponents a better position to delay the three-year ratification extension as the session neared ita scheduled early October adjournment.

Byrd said he had been informed by a source he did not identify that unless he agrees to drop ERA and two other proposals from the schedule this year (the Humphrey-Hawkins jobs bill and a hospital cost-containment bill) he would be unable to obtain time agreements on other legislation.