Hagel, who arrived in South Korea, told reporters traveling with him on his Asia tour the Defense Department and other government agencies responsible for national security will carry out their missions.
"We will fulfill our mission of maintaining the alliances we have and our troops in South Korea [and] Japan, and other treaty obligations," Hagel was quoted as saying in a release on the Defense Department website.
Hagel, however, was critical of the government shutdown, saying it casts a pall over America's credibility with its allies.
"It is nonsensical ... . It is completely irresponsible. It's needless. It didn't have to happen. And I would hope that our Congress can find a new center of gravity of responsibility, and start to govern," Hagel said.
Hagel said during a teleconference Monday night with Deputy Defense Secretary Ash Carter, comptroller Bob Hale and acting general counsel Robert Taylor, they discussed ways for minimizing the shutdown's effects on about 400,000 civilian employees who will be furloughed.
"Our uniformed military are taken care of" and will be paid, the secretary said.
Hagel said he has repeatedly been asked by South Korean officials why the shutdown occurred, adding the shutdown affects "our relationships around the world."
"Here this great republic and democracy, the United States of America, shuts down its government," he added. "The Pentagon, even though we are exempted -- our military -- has no budget. We are still living under this dark cloud of uncertainty, not knowing what's going to happen."
The secretary said a strong military is essential for national security. However, civilian employees in the Defense Department, as well as in other departments, also play a vital role in that mission.
Separately, President Obama, in a video message Tuesday to the Defense Department, said: "Those of you in uniform will remain in your normal duty status. The threats to our national security have not changed, and we need you to be ready for any contingency."
Obama said ongoing military operations, such as in Afghanistan, will continue. He assured any U.S. defense personnel serving in harm's way will "have what you need to succeed in your missions."
"To all our DOD civilians, I know the days ahead could mean more uncertainty, including possible furloughs," the president said. He said he will keep fighting to get rid of the across-the-board budget cuts -- the sequester -- which are hurting the military and the economy.