Republicans point to gains made in the U.S. House, including in some competitive suburban areas, as they say they are on sound footing for the road ahead against a Democratic Party that faces its own internal tensions and a pull to the left.

“America needs a real, multi-ethnic working class party that understands that what may be good for Wall Street and Silicon Valley may not be good for our nation’s workers, communities and families,” said Sen. Marco Rubio of Florida, a 2016 presidential contender who is expected to run again at some point.

“The Republican Party can be that party if we remain focused on winning the great power competition with China, rebuilding our nation’s industrial capacity and restoring dignified jobs,” Mr. Rubio said.