Former secretary of state Alexander M. Haig Jr., a candidate for the Republican presidential nomination, was discharged from Fairfax Hospital yesterday after being kept overnight for treatment of an allergic reaction to drugs he had taken for a tooth problem.

"They gave him a clean bill of health," said Fairfax Hospital spokesman Lon Walls.

"He went right to the airport from the hospital," said campaign spokesman Dan Mariaschin. He said Haig, 63, was traveling to a vacation home he owns in south Florida where he planned to spend the New Year's weekend with his wife and daughter.

"He'll probably play a little tennis and then come back for the stretch run," said Mariaschin, referring to Haig's return to campaigning in New Hampshire on Monday.

Haig, President Reagan's first secretary of state and a former NATO commander, had been treated Wednesday at Walter Reed Army Medical Center for an abscessed tooth and had been given penicillin and Tylenol.

Late in the afternoon, he went to the Fairfax Hospital emergency room complaining of a reaction to the medication.

Haig's face was flushed, his tongue swollen and "he became faint or lightheaded," said Thomas Barsanti, an emergency room physician.

"The reaction was to the combined use of the medications," Walls said.