"I actually think Conan knew exactly what was going on," Trump said, calling the dog "very, very special."
In October, Trump confirmed that al-Baghdadi fled from U.S. military personnel during the raid and detonated a suicide vest in a dead-end tunnel, killing himself and three children.
Conan was injured in the operation, but no U.S. personnel were killed.
Both Trump and Pence noted that they met with the special forces members, who couldn't appear publicly, but said they were happy to welcome Conan to the White House.
"I think having the special forces here today, who obviously can't come out in the public, but also having this extraordinary dog here today is all a reflection of our armed forces and the great job that they do," Pence said.
The president said the United States has done "a lot of work" combatting IS since the raid and has also received help from other nations.
"Certain things have happened that are very important. So we're in very good shape," he said.
Trump said that Conan recovered very quickly from his injuries and has since gone on "very important raids."
He also stated that Conan will not be retired following the injury, saying the dog is "primetime age-wise" to continue working.