WASHINGTON — President Obama and Congressional Democrats on Friday abandoned their once-firm stand that growing airport bottlenecks would be addressed only in a broader fix to across-the-board spending cuts, accepting bipartisan legislation that would bring the nation’s air traffic control system back up to full strength.

With remarkable speed, the House overwhelmingly approved legislation to give the secretary of transportation enough financial flexibility to shift as much as $253 million to the air traffic control system, less than a week after the onset of politically problematic flight delays driven by across-the-board spending cuts. The money will be shifted from airport improvement funds, and none would come from additional revenues, once a key demand of Mr. Obama and the Democrats. The 361-to-41 vote came less than 24 hours after the Senate rushed the measure through.

Republicans claimed victory. “Consider that the Democrats’ opening position was they would only replace the sequester with tax increases,” Representative Eric Cantor of Virginia, the House majority leader, said in a memo to members before the vote. “By last night, Senate Democrats were adopting our targeted ‘cut this, not that’ approach. This victory is in large part a result of our standing together.”

The Congressional action effectively undoes one of the thorniest results of “sequestration,” the $85 billion in spending cuts that took effect March 1 and have rippled across the federal government. With the president’s promised signature, Democrats will lose significant leverage they had hoped would force Republicans into a larger agreement since the flight delays were seen as the sort of inconvenience that could force a reversal of the cuts.