It was unclear whether U.S. or Iraqi forces were responsible for the attack Sunday.
Nine Russian diplomats, including Ambassador Vladimir Titorenko, left the Iraqi city of Fallujah, 30 miles west of Baghdad, on Monday en route to Syria, where a medically-equipped Russian plane was headed to bring them back to Moscow.
Syrian border officials, speaking on condition of anonymity, said the delegation from the Russian Embassy in Baghdad crossed the Tanef crossing point on the Syrian-Iraqi border. Tanef is 187 miles northeast of Damascus.
"They are on the Syrian side of the border," said the officials.
One diplomat injured in the attack was left behind in a Fallujah hospital, Alexander Yakovenko, the Foreign Ministry spokesman, told Russia's NTV television. Another diplomat also stayed behind to help the injured Russian, who underwent surgery in the Iraqi hospital, Yakovenko said. Others among the 23 diplomats and journalists in the convoy traveled to the Jordanian border on Sunday.
The Russian convoy was fired upon as it headed out of Baghdad toward the Syrian border, injuring at least four. A journalist in the convoy said it was caught in crossfire while passing Iraqi positions near the city's outskirts.
Alexander Minakov of state-run Rossiya television said it appeared that the U.S. forces had fired first, unleashing a heavy barrage on the Iraqi positions, and the two sides then exchanged fire.
The United States had been aware of the Russian diplomats' evacuation plans, and the convoy was flying a Russian flag.
Yakovenko did not comment on Russian media reports that bullet holes in the vehicle and a bullet removed from an injured diplomat matched the caliber of bullets from an American M-16 rifle.
He said Russia had not yet received any official information from either side, and was awaiting their conclusions.
Russian President Vladimir Putin instructed Foreign Minister Igor Ivanov on Monday to keep him informed of the convoy's movements and the status of the injured, Russian news agencies reported.
Dmitri Rogozin, head of the lower house of parliament's international affairs committee, suggested that Russia should have evacuated its diplomats from Baghdad earlier. Russia, which firmly opposes the war, had been reluctant to close its embassy, and announced the decision to evacuate the ambassador only Saturday.
The evacuation came three days after Russia alleged that American airstrikes had targeted a Baghdad neighborhood where the Russian Embassy is located.
"Clearly there was a desire to have full and objective information about developments in Iraq," Rogozin was quoted by the Interfax news agency as saying. "In this case, it would have probably been sufficient to keep a smaller group of diplomats and not leave the ambassador until the last moment, because he is a political representative of the country, after all."
In spite of Russia's opposition to the war, Putin has adopted a softer tone toward the United States in recent days, saying a U.S. defeat would not be in Russia's interests and pledging continued cooperation with Washington.
U.S. national security adviser Condoleezza Rice was in Moscow for talks Monday on deepening U.S.-Russian cooperation.