NEW YORK — Mayor Michael Bloomberg pulled off a stunning victory Thursday in getting New York City’s term-limits law changed so he can run for a third term, but the billionaire isn’t likely to start campaigning until he mends some frayed relationships.

The former CEO accomplished one of his greatest legislative wins in convincing a divided City Council to back his bill, which gives officeholders the option of running for a third consecutive four-year term.

Bloomberg’s chief argument to do it quickly, and without voter approval, was that his financial background is crucial to leading the city through the fallout from the economic crisis.

New Yorkers can expect this message to be the framework for his re-election bid when the Bloomberg campaign officially gets rolling next year, aides and associates say.