"We're trying to make sure that public transit is fully functioning in time for tomorrow's commute," he told broadcaster CBS.
Limited bus and subway service in Boston was expected to resume in the early afternoon but operators called on its customers to expect significant delays at the start of the new workweek.
Meanwhile, a spokeswoman for the National Grid power company said it was warning some of its customers in the southeast of the state that they may not have electricity restored before Tuesday.
"It's a possibility," said Charlotte McCormack, adding that some 72,000 of its customers remained without power in Rhode Island.
"We took a very major blow, but the state's responding very well," said Rhode Island Senator Jack Reed.
The storm also took a human toll.
A Massachusetts boy aged 11 died when he and his father were warming up in their car and inhaled carbon monoxide after an exhaust pipe had been blocked by snow.
A car driven by a young woman went out of control in the snow on a highway in the town of Poughkeepsie, New York, striking and killing a 74-year-old man who was walking on the shoulder of the road.
And in Auburn, New Hampshire, a man was killed after losing control of his car and hitting a tree, local officials said.
Malloy was quoted by local media as saying there were five storm-related deaths in Connecticut and, according to reports, a fatal crash in Maine claimed another life.
Minor injuries were reported in a 19-car pileup on Interstate 295 in Falmouth, Maine, caused by poor visibility and slippery road conditions.
The storm came a little over three months after Hurricane Sandy devastated swaths of New York and New Jersey, killing 132 people and causing damage worth some $71.4 billion.
As the East Coast slowly dug out, the National Weather Service warned of a new blizzard taking aim at the northern plains.
"A blizzard will continue across the north central US, where heavy snow will combine with strong gusty winds to produce dangerous whiteout conditions," it said.