Defense Secretary Leon Panetta arrived in Cairo Tuesday to meet Egypt's new Islamist president, the Muslim Brotherhood's Mohammed Morsi, and the country's top general, Field Marshal Mohammed Hussein Tantawi. After the meeting, the secretary told reporters, "It's clear that Egypt, following the revolution, is committed to putting into place a democratic government."

Such patience and reassurance is wise in the wake of an Arab Spring that brought forth democratic elections for the first time in generations. It should be remembered that the U.S. had no trouble living with the Arab autocrats. It did so, as George W. Bush once put it in a memorable speech to the National Endowment for Democracy in 2003, for six long decades—all in the name of stability. The terrible band of jihadists who struck America on 9/11 shattered that compact with the autocrats.

We are now called upon to figure out the terms of a new accommodation, and suddenly many of us are without historical patience. The Islamists in Egypt, Tunisia, Morocco have stepped forth. They didn't make that Arab Awakening of 2011 but had become its beneficiaries. Their leaders had not been on the Rolodex of Goldman Sachs, they were not clients of Washington lobbying firms. They had no favors to dispense. Suddenly history broke their way.

My generation of Arabs, who came to politics amid the high hopes of secular nationalism in the 1950s and '60s, had no use for religious politics. We didn't know these men, let alone the women in oddly stylish headscarves. For a fleeting moment, in the aftermath of the Egyptian revolution led by Col. Gamal Abdul Nasser in July 1952, the Brotherhood was confident that it would partake of the New Order. It had infiltrated the officer corps, and the men in uniform, they thought, would be their instrument in bringing about the "reign of virtue."

But the Nasserist state, and its tributaries and enthusiasts in other Arab lands, had no room for what we would now call political Islam. The Brotherhood was decimated by the Nasser regime, and in 1966, its inspirational leader, Sayyid Qutb, was sent to the gallows.

It was secular nationalism's heady moment. The modern world beckoned in all domains. There was new literature and art and culture, lively media, the promise of mighty armies and an industrial economy. But the promises of pan-Arabism were to go unfulfilled. The reckoning came in June 1967 with a defeat by Israel in the Six-Day War—days from which two generations of Arabs never quite recovered.

Yet those six days were a boon for Islamists. They didn't quarrel with fate. The deliverance that came their way was a gift of their dreaded enemy—the Jewish state that put the Arab armies to flight, and put on cruel display the fraud of so much of what their leaders had claimed. The return of the Islamists had truly begun. The broad middle classes of the Arab world were in play, their economic and psychological gains shattered. The Arabs, said the Islamists, had forsaken God, and this devastating defeat by Israel had the hand of God in it.

The intervening decades were the time of the despots—charisma quit the world of the Arabs. The rulers now had the whip. They banished politics. Mass terror made its appearance, Syria under Hafez al-Assad and Iraq under Saddam Hussein turned into slaughterhouses. In Algeria a barbarous war was fought between the Islamists and Le Pouvoir, the cabal of ruling generals.

We shall never know for certain the impact of Saddam Hussein's ouster in 2003 on the Arab revolutions that toppled dictators in Tunisia, Egypt, Libya and soon, one hopes, Syria. The spectacle of Saddam—the knight of Arabism, the self-appointed gendarme of the Persian Gulf—flushed out of his spider hole doubtless was a trauma for rulers and their accomplices, and a vicarious spectacle of liberation for those among the Arabs who yearned to see the demise of their own dictators.

The Islamist leaders with cardigan sweaters and close cropped beards were not exactly the heirs of the old Brotherhood. They had emerged out of the professional syndicates—engineers and physicians figured prominently in their ranks, as did worldly businessmen. There were those who returned from exile in the West, those with modern degrees earned in Western universities. Rachid Ghannouchi, the leader of the Tunisian Islamists, belongs to the first category, while Egypt's Mohammed Morsi, who holds a doctorate in engineering from the University of Southern California, belongs to the second.

The skeptics see them as opportunists who hijacked the democratic process. Yet what choice do we have but to accept the democratic claims of these new Islamists? We can't send the mukhabarat (secret police) after them, as past dictators did. We have to grant them time.

At any rate, the Islamists don't have the political world to themselves. In Morocco, for example, some space was created for the Islamists, and one, Abdellah Benkirane, heads the cabinet. But Morocco remains a monarchy with more than three centuries of rule behind it, where the king claims descent from the Prophet and wields religious, political and military authority.

In Tunisia, the Islamist Nahda Party received a plurality of the votes—and cut a deal with two secular parties to divide the power—and the burdens.

In Egypt, the Muslim Brotherhood has the presidency, but the Supreme Council of the Armed Forces emptied that office of much of its power. The "deep state"—the security forces, the vast apparatus of the ministry of interior—has not been dismantled.

In Libya, the Islamists contested a parliamentary election but they lost out to an older force—tribalism. The coalition that prevailed was headed by a technocrat, Mahmoud Jibril, with a doctorate in political science from the University of Pittsburgh, who was carried to power by the allegiance of his tribe and a broad "liberal" coalition.

We are not the only ones watching. Ordinary Arab men and women will be on the lookout for any cracks in the Islamists' edifice. They will look for evidence of corruption, for the possibility of sons and daughters and nephews and in-laws inheriting the new world. It will be hard for the Islamists to hide—their beards and their worry beads and their affirmations of faith shall not acquit them.

Mr. Ajami is a senior fellow at Stanford University's Hoover Institution and the author most recently of "The Syrian Rebellion," just out by Hoover Press.

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