He said: "The United States strongly believes that together with the nations of Europe, including Turkey, we must work with Azerbaijan and other countries in the Caucasus and Central Asia on additional routes for energy exports that ensure the free flow of resources."
Azerbaijan, Georgia and Ukraine, which are all former Soviet republics, are increasingly wary of Russian intentions after its conflict last month with neighbouring Georgia
Azerbaijan pumps nearly one million barrels a day of high quality crude, equivalent to about one per cent of the world's oil supplies, through a BP-led pipeline that passes through Georgia and Turkey to Europe.
However, it has said that it is re-routing some of its production to a rival route through Russia, citing the conflict in Georgia as part of the reason.
|Azerbaijan pumps nearly one million barrels a day of oil to Europe [EPA]
Ukraine and Georgia have both angered Moscow by seeking membership of the Nato military alliance.
Tbilisi, which was backed by the US during the fighting, has been weighing its next move in its relations with Russia which may include cancelling a lucrative civil nuclear deal.
A US administration official said that Washington planned to announce a package of about $1bn in aid to help rebuild Georgia on Wednesday.
Meanwhile, the US and the European Union are attempting to break Moscow's stranglehold on the transit of central Asian gas to Europe with the planned Nabucco pipeline which will pass around Russia's southern flank.
Cheney's visit to a region that Russia sees as its backyard has brought renewed attacks from the Kremlin.
Russia accuses Washington of helping to trigger the recent conflict by backing what it says is a pro-Western Georgian government bent on aggression.
Dmitry Medvedev, Russia's president, said it was time for the US to re-evaluate its policy of supporting Mikheil Saakashvili, Georgia's president.
Medvedev described the US-educated lawyer as a "political corpse" and said Moscow wanted nothing to do with him.
The president's remarks contrasted with the more conciliatory language he used about the EU, which on Monday threatened to suspend talks on a partnership pact but rejected sanctions on Russia, the bloc's biggest energy supplier.