Five weeks after rejecting a labor contract with the state, the Public Employees Federation rolled over for Gov. Cuomo, approving a deal nearly identical to the one it nixed in September.

Thursday’s vote by the members green-lighted a four-year contract with modest givebacks that create $400 million in savings for New York and plug a vast hole in the state budget.

The good news came just in time for some PEF members: Cuomo had vowed to lay off 3,500 of them if the rank and file rejected the contract.

Actually, back in September, members voted to do just that — preferring to risk job losses over minor contract concessions, like chipping in a bit more for their health insurance.

But a new round of talks led to a slightly amended deal.

The new contract appears to be little changed; it includes a three-year wage freeze, followed by a 2 percent bump in the fourth year. Had PEF voted the contract down a second time, layoffs were to go into effect yesterday.

Fortunately for them, they didn’t.

The vote passed, the layoffs were averted and Cuomo — and the state — don’t have to come up with any more money (which they don’t have).

Pretty good, all around.

The vote does raise one question, though.

In the first round of balloting, back in September, the contract went down in flames, losing in an 8-point landslide; this week, it sailed through by a whopping 40-point margin — even though it is, supposedly, virtually the same contract.

What exactly triggered the big switch?

Cuomo says cooler heads prevailed, and union members feared layoffs.

That may well be true.

But then, it wouldn’t be the first time in New York state politics that certain winks and nods — regarding, perhaps, hidden promises — made all the difference.

And the nearly 20 million New Yorkers who aren’t PEF members are in no mood for expensive surprises.

Let’s hope there aren’t any.