Still there will most be a push for even more: There is already chatter about legalizing recreational marijuana; investing more in the crumbling subways; addressing climate change; and installing an early voting system for the first time, all ideas that the governor has previously offered differing degrees of support for.

And after a tumultuous primary season in which liberal challengers ousted seven more moderate Democratic state senators, the virtual Holy Grail of current progressive thought seems within reach: a state-run universal health care system. Mr. Cuomo stopped short of fully embracing the proposal, saying it “has merit.”

Andrea Stewart-Cousins, who is slated to become the first woman and first female African-American to lead the New York State Senate, also struck a note of caution about the Democratic agenda. “You eat the elephant one issue at a time,” she said on Tuesday, as votes began to roll in for her party.

There also will likely be, of course, intraparty differences. The Senate that Ms. Stewart-Cousins will lead is very likely to contain several factions, including New York City liberals and more moderate members from the city’s suburbs. Fifteen of her members will be new to Albany, and its vagaries, with many of that cohort never before having held public office — or, in some cases, even having an office.

[What drove the rogue group of Senate Democrats to work with Republicans? Read more here.]

Gustavo Rivera, a Democratic state senator from the Bronx, said that Democrats had failed to keep the trust of voters during their brief time in charge nearly a decade ago and know they must do better this time.

“Governing is a very complicated matter,” Mr. Rivera said. “It’s not just giving raises to your staff and getting nicer offices.”