"Once you're brought to attention by a foreign government, I think you should have a red flag put then, to be taken off later," Graham said.
But on Monday, Graham said that after a discussion with an assistant director at the FBI, he believed that instead "we need to revisit our laws."
"The FBI either dropped the ball or our system doesn't allow the FBI to follow this guy in an appropriate fashion," Graham said. "I think, once the Russians made the request, the FBI did a good job of looking at him. The reason we didn't know he went to Russia is because the name was misspelled."
Graham said he did not know if Tsarnaev may have intentionally misspelled his name to avoid detection.
"They say that Aeroflot gave us the information, but could he have done it? Yes. I don't know. What the FBI told me sounded very reasonable," Graham said.
Tsarnaev, 26, died in a shootout with police early Friday morning following a police pursuit through suburban Boston. His 19-year-old brother, Dzhokhar, escaped from the shootout but was later discovered hiding in a nearby boat, where police arrested him.
Graham said that while he didn't want "a police state," he wanted to adjust laws to create "a nation where the police can protect us."
"The sooner we realize that we're at a war with radical Islam and come up with systems to defend ourselves, the better off we'll be," Graham said.
The South Carolina lawmaker also reiterated his desire for the younger Tsarnaev brother to be classified as an enemy combatant.
"This idea, the only way we can question him about national security matters is to go through his lawyer … that is absolutely crazy," Graham said. " This man should be held and questioned under the law of war. The information can't be used against him, but it can be used to help us and citizenship doesn't give you immunity from the law of war."