So the Senate and House just passed the bill extending the payroll tax cut and unemployment benefits for the rest of the year, capping the end of a months long fight that did significant damage to the GOP. The compromise is a victory for the White House; 160 million working Americans will enjoy a continued tax cut, and millions more will see extended jobless benefits, which in theory should help speed the recovery.

But the deal is an imperfect one, and questions linger among labor officials and some Democrats over whether the White House and Dems conceded too much ground towards austerity and requiring sacrifices from workers.

Republicans, too, deserve credit for reaching this deal. This will allow them to move past a fight that had showcased divisions in the party, muddled their claims of fiscal responsibility and reinforced the Dems’ chosen framing of the coming election — that GOP priorities favor the rich at the expense of the middle class.

Indeed, the end to this battle showed how much Republicans wanted to put it behind them. Senate Republicans agreed today to a simple majority-vote threshold. House Republicans agreed to extend the payroll tax cut without paying for it with massive spending cuts that would have been difficult for Dems to swallow and would have harmed the recovery.

That was the biggest concession of the whole battle. But it also allowed Republicans to separate the payroll tax cut from unemployment benefits, restoring some of their leverage. Republicans got the length of the unemployment extension chopped down, though Dems note the deal retains a longer extension in high unemployment states and nixed onerous conditions on the jobless.

But unemployment benefits were reduced. And the deal has left labor officials and some Democrats angry, because the unemployment extension will be paid for in part by an increase in federal employee contributions to their pension plans — even as Dems dropped their demand for a millionaire tax. One labor official noted to me that the Republicans’ desire to get this behind them showed that they had a weak hand and that Dems could have extracted more by maximizing their leverage. This was a win for Dems, but with one foot on GOP/austerity turf.

Still, Obama has now successfully pressed Congress to pass a major piece of his jobs agenda at a moment when the Republican presidential candidates are facing weeks of damaging political squabbles about issues that don’t matter to swing voters. The trajectory of this fight will embolden the White House to continue aggressively calling out GOP obstructionism and denouncing misguided Republican priorities. The win, coming amid rising approval numbers for Obama, could give him still more upward momentum. That could in turn make it tougher still for the GOP — whose popularity is sinking even as Obama is getting more credit for the recovery — to stand in the way as he prods Congress to keep moving the economy forward.