The almost 900-foot-long cruise ship pulled away from the Alabama Cruise Terminal slowly and began the 29-mile trip down Mobile Bay to the Gulf of Mexico.
The company said the ship would head east to a dry dock in Freeport, Bahamas, for more repairs and upgrades.
Triumph is due to arrive at Freeport on Saturday, and it is scheduled to return to service June 13 with a cruise from Galveston, Texas.
The Triumph was towed into Mobile on Feb. 14 after a Gulf cruise turned into a nightmare for passengers who spent five days in what many described as squalid conditions, with some floors awash in human waste.
In April, as the ship underwent repairs at BAE Systems, it broke loose from its moorings during a storm and was damaged further. A dock worker died after being swept into the water.
The manager of the cruise terminal, Sheila Gurganus, said repairs continued at the wharf after the mishap.
"They're done a lot of work here," she said. "They've been running cable, and they'll continue working on it while it's under way."
Meanwhile, Carnival Corp. is seeking the dismissal of lawsuits filed by passengers. The biggest is a potential class-action lawsuit seeking to represent about 3,000 passengers. Carnival said in a motion filed last month in federal court in Miami that its cruise tickets clearly state passengers cannot file class-action lawsuits.
Lawyers for the passengers say the lawsuits should be allowed because Carnival was negligent in letting the Triumph sail with a history of mechanical problems.