Tuesday's vote followed a petition with more than 120,000 signatures of people urging lawmakers to approve the measure.
The American Center for Law and Justice has been following Nadarkhani's case and advocating on his behalf.
ACLJ Chief Counsel Jordan Sekulow said he is "hopeful" that the entire Senate will propel the resolution forward.
"Pastor Youcef Nadarkhani has sacrificed so much for his faith and has refused his captors' calls for him to renounce Christ," Sekulow said in a message to supporters.
He added that the resolution is key to strengthening the international fight against religious persecution toward Nadarkhani and others.
The ACLJ is now pressing Senate Majority Leader Harry Reid to call a full vote on the issue.
Nadarkhani is facing death by hanging in Iran for refusing to recant his Christian faith. He's been in prison away from his wife and two sons since 2009, often under harsh treatment and poor living conditions.