It took a war to recognize it, but Iraq is not the key to meeting U.S. goals in the modern Middle East. That distinction goes to Iran. Achieving stability in Baghdad and Kabul, guaranteeing the safe passage of Persian Gulf oil, securing an Israeli-Palestinian peace agreement, salvaging Lebanon’s democracy, and pushing Syria toward more cooperative policies—all these American objectives have a better chance of being met if Tehran has a place at the table.

Yet the U.S. approach toward the Islamic Republic remains ossified. The mullahs in Tehran continue to be branded as nuclear-obsessed terrorists, treated as international pariahs who only understand threats and isolation. The White House’s talk of World War III reveals a fundamental U.S. error: Iran’s policy tools—the threat of a nuclear weapons program, its ties to Hezbollah and Hamas—are not its policy goals. What the mullahs crave is not nuclear suicide but a legitimate regional role, and they are determined to achieve that influence, with or without American blessing.

That’s why the next U.S. president should seize the upper hand and embark on a “Nixon in China”-like visit to Tehran. Thirty-five years ago, Richard Nixon and Henry Kissinger sat down with Mao Zedong and transformed mutual enmity into a quasi alliance. That American enterprise scored a Cold War victory against Moscow. The next U.S. president would accomplish a similar strike against al Qaeda and the forces of instability in the Middle East, while guaranteeing that Iran’s nuclear plans remain only on the drawing board.

Unlike most states in the region, Iran was not born this century. It is the world’s second-oldest state after China. Regimes have come and gone in its long history, and change...