The Emmys’ big winner among networks was a traditional powerhouse, the premium-cable channel HBO, even in a year without “Game of Thrones.” The limited series “Big Little Lies” and “The Night Of,” the comedy “Veep” and the late-night show “Last Week Tonight With John Oliver” all won multiple awards.

Many of the night’s other awards went to newer peak-TV voices. Donald Glover won directing and acting awards for his distinctive comedy “Atlanta” on the cable channel FX. Netflix’s anthology series “Black Mirror” won two awards, including best TV movie for its episode “San Junipero.” Aziz Ansari and Lena Waithe won a writing award for the Netflix comedy “Master of None.”

And the night’s triumphant show was another series from a streaming service, Hulu’s “The Handmaid’s Tale,” which took home best drama and best actress among its five awards on the night. (That show’s Elisabeth Moss won for the first time after multiple nominations for “Mad Men.”)

Direct jabs at Mr. Trump were not infrequent, though it was perhaps surprising how many winners declined the opportunity to make any sort of reference to the national mood. Mr. Glover credited Mr. Trump with his award for actor in a comedy, saying he had made “black people No. 1 on the most-oppressed list.”

Julia Louis-Dreyfus, accepting her sixth comedy-actress award for “Veep,” joked that the show had considered an impeachment story line for its final season but abandoned it “because we were afraid someone else might get to it first.”

The broadcast went off with few hitches, though more winners than usual had to be played off the stage to meet the 11 p.m. Eastern ending time. The 92-year-old Cicely Tyson, presenting the award for outstanding limited series, stumbled at first but rallied strongly, to fierce applause. Celebrating the industry’s inclusiveness, Hayma Washington, president of the Television Academy, made a possible Freudian slip when he cited its “insclusiveness.”

One of the night’s more touching moments celebrated a pioneering woman in the TV industry and a CBS hero: The In Memoriam segment ended with Mary Tyler Moore, who died in January at 80, turning off the lights in the fictional newsroom of “The Mary Tyler Moore Show.” It had nothing to do with national politics, but it felt like an apt comment on current trends in the broadcast TV business.