WASHINGTON — President Obama shook off an electoral drubbing on Wednesday and said he was eager to find common ground with Republicans during the final two years of his presidency, but he swiftly defied their objections by vowing to bypass Congress and use his executive authority to change the nation’s immigration system.

In a sign of how he intends to govern under a new political order with ascendant Republican leaders, Mr. Obama renewed his commitment to act on his own to allow millions of undocumented immigrants to stay in the country.

His remarks, at a news conference in the East Room of the White House, were meant to put the vitriol of the campaign behind him — he responded to disaffected Americans by saying that “I hear you” and that his election mandate was to “get stuff done.” But his promised action on immigration underscored the profound partisan disagreements that persist in Washington.