Republicans who have shied from the responsibility of government will now be called upon to support increases in the debt limit, approve annual budgets, endorse spending bills and back other must-pass measures that they formerly left to the Democrats and some of their more compromising colleagues. With Democrats unlikely to help on many of those votes after being castigated for them by Republicans, the Republicans who belonged to the “vote no, hope yes” caucus when it came to critical legislation in recent years now will have to vote yes and hope things go well.

This isn’t the same style of Republican majority pushed from power after being routed in the 2006 midterm elections after the public backlash to the administration of President George W. Bush and his handling of the war in Iraq.

Forged by the Tea Party revolt that restored Republicans to control of the House in the 2010 elections, and in the Senate in 2014, this party is much more conservative with a membership that tends to see government as an impediment to be leveled, not as a force to be shaped to their views to the benefit of their constituents. Eight years of railing against the Obama administration has infused them and their constituents with a hostility and disregard for the government that Republicans must now lead rather than ridicule.

Tensions could arise between House and Senate Republicans as well. When the Newt Gingrich-led party took over the House in 1995 for the first time in four decades, newly empowered Republicans sent a raft of legislation to the Senate, only to see it stall there. With President Bill Clinton in the White House at the time, Republicans knew much of it would not be enacted. Now, with Mr. Trump soon to occupy the Oval Office, it is unlikely that House Republicans will be willing to watch Democrats bottle up legislation in the Senate. Demands that their Senate counterparts eliminate the filibuster could mount quickly.

While they understand the challenges, Republicans are nonetheless jubilant at their enviable position.

“A Republican in the White House and a Republican majority in Congress present tremendous opportunity to make real progress,” Senator Cory Gardner, Republican of Colorado, said in the party’s weekly radio address on Saturday. “We assume that responsibility with the promise that we’ll work hard to do everything that we can to deliver more opportunities to Americans tomorrow than they have today.”

“I am pretty giddy,” said Mr. Cole as he looked ahead.

Republicans have won their chance. Now it is time to see what they can do with it.