WASHINGTON — President Trump’s cabinet continued to take shape on Monday, as Representative Mike Pompeo of Kansas earned approval to lead the Central Intelligence Agency and Rex W. Tillerson, the secretary of state nominee, cleared a key Senate hurdle to all but assure his own confirmation.

Despite some shaky appearances from his nominees on Capitol Hill and often blistering Democratic opposition, Mr. Trump has thus far faced few meaningful obstacles in installing the team of his choice — aided by Senate Republicans who are eager to expedite the confirmation process.

Mr. Tillerson had been the only nominee to attract notable doubts from within the Republican Senate majority, briefly imperiling his prospects amid concerns over his ties to Russia and its president, Vladimir V. Putin, during Mr. Tillerson’s tenure at Exxon Mobil. But on Monday, Senator Marco Rubio of Florida, who had emerged as a critical vote on Mr. Tillerson’s fate, announced his support.

Mr. Rubio had sharply questioned Mr. Tillerson at a hearing this month, chastising him in a series of interrogations over Mr. Tillerson’s equivocating testimony on human rights issues. Mr. Rubio also had left open the possibility of opposing him on the Senate Foreign Relations Committee, where Republicans hold a one-member majority.