Johnson, who turns 60 on Dec. 28, was admitted to George Washington University Hospital with an undiagnosed illness, said a spokeswoman, Julianne Fisher. However, she said the senator did not suffer a stroke or heart attack. His office had said earlier Johnson may have had a stroke.
There has been no word on the nature of Johnson's surgery, which lasted past midnight, or on his condition.
Sources close to the situation, speaking before the surgery was made public, tell CBS News the situation is definitely not good.
Johnson became disoriented during a call with reporters at midday, stuttering in response to a question. He appeared to recover, asking if there were any additional questions before ending the call.
"Every member of the United States Senate sends our best to him and to his family at this difficult time, and we wish him a full recovery," said Senate Majority Leader Harry Reid, who calls Johnson "a dear friend."
Johnson's condition could determine control of the Senate, CBS News correspondent Gloria Borger reports.
Should Johnson be unable to continue to serve, it could halt the scheduled Democratic takeover of the Senate. Democrats won a 51-49 majority in the November election.
South Dakota state law says that the governor can appoint a replacement when there is a "vacancy," but Secretary of State Chris Nelson tells CBS News political producer Steve Chaggaris that it's unclear whether a seat held by an incapacitated senator would be considered "vacant."
Unless Johnson dies, the state of South Dakota is unsure of how to proceed if he's incapacitated. If he dies, however, the governor, Mike Rounds, would appoint a replacement who would serve until the next general election in 2008, Chaggaris explains.
Because Rounds is a Republican, an appointment of this nature could affect the balance of power in the Senate, and thus the entire Congress. Leadership would be given back to the Republicans, because in a split Senate, the deciding vote is given to the vice president.
Rounds released a statement saying, "we are hopeful of good news for our friend and colleague."
Johnson, 59, a fourth-generation South Dakotan, was born in Canton to college professor Van Johnson and homemaker Ruth Johnson. He attended school in Canton, Flandreau and Vermillion and graduated from Vermillion High School in 1965.
He attended the University of South Dakota, where he met the woman he would marry shortly after graduation — Barbara Brooks of Sioux Falls. Johnson went on to earn both a master's degree in public administration and a law degree from USD.
Johnson started a law practice in Vermillion in 1975 and ran for statewide office in 1978. He served four years in the state House of Representatives and another four years in the state Senate before setting his sights on Washington.
He was elected to South Dakota's U.S. House seat in 1986 and served five terms before moving to the U.S. Senate in 1996.
Johnson was most recently re-elected in 2002, narrowly defeating then-Rep. John Thune, and currently serves on the Senate's Appropriations Committee, Budget Committee, Banking Committee, Energy and Natural Resources Committee and the Indian Affairs Committee.
The couple, who still have a home in Vermillion, have two sons and a daughter: Brooks, who served in the U.S. Army in Bosnia, Kosovo, South Korea, Afghanistan and Iraq and is still in the military, posted stateside at the moment; Brendan, a Sioux Falls lawyer; and Kelsey, who works in public service in Washington.
Both Johnson and his wife have battled cancer.
The senator underwent prostate cancer in 2004 and subsequent tests have shown him to be clear of the disease. Barb Johnson is a two-time breast cancer survivor.