Obama reassures supporters in 'fiscal-cliff' video

"Just recently, Republicans in Congress said they'd never agree to raise tax rates on the wealthiest Americans. We've now raised those rates permanently, making our tax code more progressive than it's been in decades," Obama said in the video, posted to Obama's Facebook page.

The video comes as the president was criticized by some on the left for giving up too much in the deal. Under the legislation, which was approved in the Senate and House on Tuesday, Bush-era tax cuts were preserved for incomes under $400,000 — higher than the $250,000 initially requested by the president. Dramatic cuts to defense and domestic spending were postponed for two months, and no compromise to avoid another battle over the debt ceiling was reached — both points on which the White House had fought.

“Every dollar that wealthy taxpayers do not pay under this deal, we will eventually ask Americans of modest means to forgo in Social Security, Medicare and Medicaid benefits," Sen. Tom Harkin (D-Iowa) said in a statement. "It is shortsighted to look at these issues in isolation from one another, especially when congressional Republicans have been crystal-clear that they intend to seek spending cuts to programs like Social Security just two months from now, using the debt limit as leverage.”

Liberal groups and unions including the Progressive Change Campaign Committee and the AFL-CIO also urged Democrats to vote against the legislation, saying the deal did not lock in significant enough tax increases.

But in his video Obama emphasized the extension of unemployment insurance, along with the continuation of tuition and clean energy tax credits, as wins for progressives. He also credited supporters with advocating for the framework of the tax deal.

"Thanks to so many of you — because you made your voices heard throughout this debate — we've stopped this middle-class tax hike," Obama said.

Still, the president acknowledged that he had preferred a grand bargain rather than "solving this problem in several steps."

"I'll admit, that hasn't always been as easy as I'd hope it would be," Obama said.

He also asked supporters to continue lobbying Congress as new battles over the debt ceiling and sequester cuts come in the new year.

"Winning an election won't bring about the change we seek on its own, it only gives us that chance to make that change," Obama said.