A rescue operation got under way to save the men after Dover coastguard received an alert at about 09:30 BST to the three-metre white dinghy off Hastings.
Lifeboat crews said the alert came after the dinghy started to leak.
The Hastings lifeboat and Kent-based coastguard helicopter were sent, but a Border Force cutter picked the men up.
A Home Office spokeswoman said the coastguard contacted the Border Force with a report that three individuals in a small boat required assistance.
She said: "HMC Seeker was deployed immediately and has picked up the individuals onboard who are believed to be Iranian nationals. The group will now be processed."
The men were brought back to the Port of Dover.
The Maritime and Coastguard Agency (MCA) said its officers were called to an incident off the Kent/Sussex coast to rescue three men from a boat in trouble.
In a statement, the MCA said: "The boat has now now been located and the incident has been handed over to UK Border Force."
Three men, thought to be Iranian nationals, have been rescued from a sinking vessel off the Sussex coast.