New York Democrats said early Wednesday that they believed they would take control of the State Senate — currently the last significant power base of Republicans in Albany — after Democrats won several critical legislative races on Tuesday.

Republicans did not concede defeat, and Albany watchers said it could be days before either party is able to assert control of the Senate, because the Democratic caucus will include five members who have not ruled out backing Republican leadership of the chamber.

Republicans are eager to keep a foothold in Albany, where Democrats, who now outnumber Republicans two to one among the state’s electorate, hold all statewide offices and control the Assembly. Control of the Senate will likely determine several major public-policy issues: the Senate Republican majority has helped Gov. Andrew M. Cuomo, a Democrat, restrict the growth of property taxes and rein in state spending, but has also prevented him from raising the minimum wage, overhauling the state’s campaign-finance system and tightening gun laws.

Republicans currently hold 33 of the 62 seats in the Senate. The chamber will expand to 63 seats next year as part of a redistricting plan drawn up by Republicans in a bid to increase their chances at keeping power.