A statement on Carnival's website said nobody was hurt and automatic extinguishers put out the fire before it could spread from the aft engine room. The company said the US Coast Guard had been notified and was monitoring the situation.
The cruise line said another of its ships, the Carnival Elation, responded to the scene and transferred provisions to the disabled boat. Carnival says a tugboat will tow the Triumph to Progreso, Mexico, where passengers will be flown back to the United States.
The US Coast Guard says tugboats are scheduled to reach Triumph about noon Monday some 150 miles off the Yucatan Peninsula. The ship is expected to arrive in Progreso on Wednesday.
Carnival says all passengers will get a full refund, plus an equal credit for a future cruise. That's a fairly standard industry response. In fact, it's what Carnival did the last time one of its cruise ships was towed back to port in November 2010. Then, as now, an engine fire disabled Carnival's cruise ship Splendor just 14 hours into a seven-day cruise.
"So as a cruise ship company, what do you offer your 3,299 customers who have gone more than two days with no air conditioning, electricity, hot running water, hot food service, or telephones, who have limited running water and bathroom facilities, and who are being towed back to port by tugboats at an average four miles per hour?
Carnival has promised a full refund for this trip, as well as a free future cruise of equal value," reported The Christian Science Monitor in November 2012.
The Triumph departed Galveston on Feb. 7 and was due to return Feb. 11.
The company also canceled the Triumph's next two voyages, set to depart Feb. 11 and Feb. 16, and is offering a full refund and a discount on a future cruise.