Mayor Bill DeBlasio's office early on reported the outage was as a result of a "manhole fire".
It said the outage extended from Fifth Avenue west to the Hudson River, and from the west 40s north to 72nd Street.
Con Edison said it was working to restore power to 42,000 customers, mostly on the west side of Manhattan.
Forty-two years ago to the day, on July 13 1977 some 9 million were affected by a massive power outage in New York City.
"Several stations are now without power and are being bypassed by all trains", it posted on Twitter.
The Fire Department of NY said it was responding to multiple calls of people stuck in elevators. "We are responding to extensive outages on the Westside of Manhattan". Several cast members from the show put on an impromptu performance in the street outside the theater for disappointed audience members. We will continue to provide updates as soon as we have more information.
Power went out early Saturday evening at much of Rockefeller Center and reached the Upper West Side, according to the ConEd Outage Map.
A diner on Broadway at West 69th Street lost its lights, as did other surrounding businesses. He went outside and saw traffic and street lights were out.
"From what we're seeing at this moment, this is simply a mechanical problem and one, again, that sounds like it is addressable in a reasonable period of time", de Blasio said.
New York City Council Speaker Corey Johnson tweeted that Con Edison's CEO told him there was a "a major disturbance" at West 49th Street Substation. At first she thought something happened at Trump International Hotel and Tower, she said.
Posts to social media showed landmarks in the city without power, including Rockefeller Center, the Empire State building and parts of Times Square. And the transit officials warned people to "avoid below-ground subway stations".
Police officers waved people out of the Columbus Circle subway station around 7:30 p.m.
While trains are still running despite the outage, the stations without power are being bypassed by trains until power is restored.