The route's opening reflects increasing ties among the three anti-American states.
Relations between Iran and Venezuela have been boosted under presidents Hugo Chavez and Mahmoud Ahmadinejad and Syria is seen as Iran's closest ally in the Arab world.
Venezuela also has a significant Syrian community.
Badr said he hoped the route would help "in decreasing harassment that Syrian, Arab and Muslim travellers are subjected to when they use other airports."
Last month, Iran Air opened an office at the headquarters of Conviasa, Venezuela's state airline, in Caracas.
Conviasa is expected to soon open an office in Iran, the Venezuelan government has said.
In January, Iran and Venezuela announced a joint $2bn fund to finance investments in their countries, as well as projects in other nations seeking to help thwart US influence.
The two leaders have also spoken of investing in infrastructure, social and energy projects.