Washington ? The nation’s unemployment rate unexpectedly fell in May for the first time in three months, to 5.8 percent. Economists cautioned the improvement probably would be temporary and new jobs would be hard to find.

The Labor Department also reported Friday that companies had added 41,000 new jobs last month after a revised level of 6,000 in April. That’s the first time payrolls have increased in two consecutive months since March 2001, when last year’s recession started.

“Ultimately it is jobs that matter, and here we see that the labor market continues to lag,” said Joel L. Naroff, president and chief economist of Naroff Economic Advisors. “Payrolls are rising, but very slowly.”

May’s jobless rate fell 0.2 percentage point from a nearly eight-year high of 6 percent in April.

“I think it’s a fluke,” said Mark Zandi, an economist. “We’ll see higher unemployment before we’ll see sustained lower unemployment.”