Dozens walked off the floor at one point, forcing the House to adjourn for the night.
The Legislative Council is scheduled to adjourn for the summer after considering the security legislation.
But the Senate said it would finish its business before daybreak and adjourn for the summer.
The legislature, set to adjourn for the year, will most likely hold a final vote on Saturday.
The bill was adopted during a flurry of activity as lawmakers rushed to adjourn for the year.
The Senate, having finished addressing all its other major issues, announced plans earlier to adjourn for the year this week.
In a radio interview, Mr. Cuomo said he did not expect the lawmakers to approve either measure before Thursday, when they are scheduled to adjourn for the year.
The outcome remains uncertain with just a week left before the Legislature is scheduled to adjourn for the year, and as pressure builds for a vote within days.
The House and the Senate were in session this weekend as lawmakers attempted to complete work on several measures in time to adjourn for the year before Thanksgiving.
Many moderates, including Representative Jim Kolbe of Arizona, urged their colleagues to wrap up the remaining issues quickly and adjourn for the year.
After passing the bill, the Senate Republicans hastily dispatched it to the Assembly and filed out at daybreak to adjourn for the summer.
