A temporary substation will provide limited electricity to power the trains. Other trains are running on diesel.
Metro-North commuters will face another week of long delays.
Power will be fully restored to the New Haven Line on Oct. 7, but it was unclear if Metro North Service would be back to normal on Oct. 7 or Oct. 8, according to Gov. Dannel Malloy.
Monday morning was the first time any electric trains have been able to run through an eight-mile section of track between Harrison and Mount Vernon after a Con Edison feeder cable failed last week.
A temporary substation is providing limited electricity to the line, powering two trains at once. Other trains are running on diesel. This will provide for about 50 percent of normal traffic, according to officials.
As of 8:30 a.m. on Monday, Metro-North's New Haven Line had carried about 13,400 passengers. The trains normally carry about 17,900 passengers, according to Malloy.
In addition, more than 8,000 park-and-ride spaces have been created in Westchester County and the Bronx so riders can reach other MTA services into Manhattan. New York City Transit officials have organized 72 shuttle buses.
"We also saw an uptick on the Harlem branch of about 2,500. We also know about a thousand people used the bus system as well," Malloy said at a news conference on Monday.
Malloy said he spoke to MTA chairman Tom Pendergast and was confident Metro North weekly or monthly ticket holders would receive some type of reimbursement.
"I suspect what it will ultimately be is a credit towards future purchase," Malloy said.
On Sunday Sens. Charles Schumer, of New York, and Richard Blumenthal, of Connecticut, announced they have sent a letter to officials at the Department of Energy and New York's Public Service Commission, asking them to help restore power and figure out what caused the Wednesday power outage at a suburban New York substation of the Metro-North Railroad.
"One reason we want a federal investigation: we don't want Abbott and Costello," said Schumer. "We don't want Metro-North blaming Con Ed, and Con Ed blaming Metro-North, and both of them blaming Amtrak, and then they're altogether blaming something else."
MTA officials said they would cooperate with investigators.
"We’re going to work very aggressively and very cooperatively with anyone doing an investigation in terms to finding out what happened, so we can learn from that and make sure it never happens again,” said MTA chairman and CEO Tom Prendergest at a Sunday evening news conference.
MTA officials said on Sunday that they're working to reimburse New Haven Line customers.
"I will be making a strong recommendation to the board, as its chairman and CEO, to consider some form or re-enumeration, either in the form of a refund or a credit, to soften those impacts," said Prendergest.
Officials also encouraged commuters to use the Hudson line when possible, as the Harlem line has been especially congested since the power failure.
Some Amtrak trains were also affected by the outage.
Amtrak's Acela service was shut down from New York to Boston this weekend and will resume with limited capacity on Monday.
New England Regional service will run on diesel power, according to Amtrak.