WASHINGTON — The furlough of air traffic controllers delayed more than 1,200 flights on Monday, the first weekday of the unpaid leaves, the Federal Aviation Administration said Tuesday as lawmakers criticized the agency for how it was handling the automatic budget cuts.

Apart from the furloughs, the F.A.A. counted more than 1,400 delays for weather and other reasons. On Tuesday, it said there were delays in the New York area and in Washington because of “staffing challenges,” but it did not say what fraction of the delays were due to furloughs.

“Travelers can expect to see a wide range of delays that will change throughout the day depending on staffing and weather-related issues,” the agency said in a statement. The staffing problems are mostly in radar rooms, not control towers, it added.

There are 30,000 to 35,000 commercial flights a day in the United States. Some travelers were delayed a few minutes, but others missed connections and were delayed for hours. Some later flights were delayed or canceled because the airplanes they were to use had arrived so late.