After months of searching for an elusive "victory" to establish President Carter as a strong leader, White House officials reacted with a mixture of relief and euphoria to yesterday's Senate vote approving the first of the Panama Canal treaties.

Officially, Carter aides predicted that gaining approval of the second treaty will be as tough, perhaps tougher, than the battle that led up to yesterday's climatic vote.

But there was no denying the signs - from the round of congratulations outside the office of Hamilton Jordan to the yelp of joy from a press office secretary as the final votes were tallied - that the Carter aides believed the president had won an important victory and, perhaps more important, averted a potentially debilitating defeat.

Shortly after the vote, Carter, smiling broadly, appeared before the television cameras in the White House press room to praise the Senate leadership, former president Ford and other "Democratic and Republican leaders" for their help in winning Senate approval of the treaties.

"The vote today is, of course, only the first step in the process of ratification, but I am confident that the Senate will show the same courage and foresight when it considers the second treaty," the president said. "This is a promising step toward a new era in our relationships with Panama and all of Latin America."

Privately, White House officials said that there were three senators, in addition to the 68 who voted for the treaty, who had promised the administration their votes "if needed." One of these, it was leaned, was Sen. Jennings Randolph (D.W.Va), who waited until approval of the treaties was assured to vote "no."

And while White House aides prepared for an equally rough battle over the second treaty, there was an underlying confidence that those senators who stuck with the administration through the first vote are not likely to switch sides a few weeks from now.

"It's going to be awfully hard to budge them," one Carter aide said. "I must say you'd have to be a masochist to put yourself in that position."

For the president, the most important factor in yesterday's vote may have been simply the avoidance of defeat.

It was six months ago that White House aides began openly admitting that they badly needed a highly visiable victory to prevent further erosion in Carter's popular support. They made a major push on the president's national energy legislation, but as the months passed and the energy legislation remained bogged down in the Senate, the erosion continued and accelerated.

With the president's full prestige and months of intensive White House lobbying behind the canal treaties, a loss yesterday would have been a severe blow to Carter's credibility as a national political leader.

"Politically, it was obviously very important," one White House official said of the long-term stakes for Carter's presidency in yesterday's vote.

The lobbying went on up to the last moment, and included telephone calls by the president Sunday to 16 Senators who were considered, even by the White House, to be against the treaties.

"I thought he had gone crazy," one senior aide said of this tactic. But the aide also claimed that in the Sunday calls the president switched one vote and got one of the three "if-needed" commitments.

Asked in the briefing room "how it feels to win one," the president said, "It feels good."

Aside from Carter, the big winners in yesterday's vote were the majority and minority leaders of the Senate, Robert C. Byrd (D.W.Va.) and Howard H. Baker Jr. (R.Tenn.).

Byrd, in his second year as Democratic leader, has been the subject of some criticism from the White House and others for his reluctance to crack the whip on the president's behalf.

But he orchestrated the handling of the Panama treaties with a skill that won commendation from administration officials. Byrd delayed his own endorsement of the treaties until he judged that the political climate permitted their favorable consideration. He then allowed hesitant senators to gain political credit from their constituents for winning one concession after another from the administration as a condition of their support.

Baker's role entailed even greater political risks, for the treaties had been a divisive issue in the Republican Party ever since Ronald Reagan hit on them as an issue in his almostsuccessful 1976 challenge to Ford.

Conservative elements in the GOP warned that anyone who backed the treaties would rule himself out of consideration for the 1980 Republican nomination. But Baker gained big personal publicity by concessions he won during a trip to Panama and then skillfully worked the Republican ranks to bring company to his position.

In the end 15 Republican senators supported the treaty, while 21 opposed it - a split that was close enough to make Baker anything but a pariah in his own party.

What was true of Carter was equally true of Baker, however. Averting defeat was mort important than any tangible political gain from winning. Baker still faces a major struggle for the nomination, and he knows that two possible rivals - Sen. Bob Dole (R-Kan.) and Sen. Richard P. Schweiker (R-Pa.) - took the opposite side on the treaty question.

Baker's vote was hissed by some 350 conservative activists listening to a radio broadcast of the roll call during a conference at the Hyatt Regency Hotel.

Later, Rep. Philip M. Crane (R-III.) chairman of the American Conservative Union, one of the sponsoring groups of the fifth annual Conservative Political Action Conference, said the defeat may serve as a catalyst for the conservative cause.

He said the conservative movement also has been able to expand its financial base through its opposition to the canal treaties, and that in the last year the ACU has benn able to increase its list of contributors from 195,000 to 300,000 names.