For the first twentysome minutes of the first episode of his new half-hour HBO show, John Oliver seemed in danger of bombing. Then he trapped General Keith Alexander, the recently retired head of the National Security Agency, on camera, and made him sit through a series of NSA rebranding suggestions that included a kitten called Mr Tiggles and, simply, “the Washington Redskins”.

The host gleefully turned the screws on the spy. The general soldiered on. The live studio audience, watching the segment on tape, roared and applauded.

Last Week Tonight with John Oliver is the future of John Oliver, if there is to be one. The former correspondent for The Daily Show with Jon Stewart won laudatory reviews last summer when he subbed in the host’s chair. Now Oliver is trying to convert his popularity into bigger stardom, a half-hour at a time, every Sunday night for a contractual minimum of two years.

Judging by the first installment of Last Week (tagline: “Breaking news… on a weekly basis”), the host might do better the more he gets out from behind his hard-won studio desk.

Oliver, 37, began his career as a stand-up comic and was hired on the strength of an audition tape about eight years ago to become a Daily Show writer. He quickly became an audience favorite for his uncanny ability to greet the most glorious examples of on-camera thoughtlessness with a thoughtfully straight face. As a roving correspondent with an “exaggerated” (they tell us) British accent, Oliver interviewed gun maniacs and Tea Party maniacs and Occupy maniacs and many other examples of the indigenous American berserk. Occasionally he broke his straight face to play his other strongest card as a comedian: outraged disbelief. As he tells the gun guy: “Are you fucking kidding me?”

Oliver’s new show is kind of like his old show, except he’s not in it as much. Which is to say that while Stewart, from the Daily Show captain’s chair, could break up his riffs on politics and the media by pitching it to John Oliver, John Oliver apparently has no one to pitch it to, and so he ends up talking too fast and too much and sounding not quite like the himself we know.

On Sunday, Oliver opened with a Daily Show-style recap of wack and wacky news that, for its scattered laugh lines, kept feeling like it was about to end but would not. He did Cliven Bundy. He did the Vatican. He did Oregon wasting almost $250m to build a health care website that never worked (first applause line of the night: “That has got to be a bitter pill to swallow for the people of Oregon. Or it would be, if they could get the pill, which they can’t, because their shitty website is broken”). He did presidential politics. He did Indian politics.

Indian politics? In fact, Oliver invested a daring five-minutes-plus on the parliamentary elections in India, which, with 850m eligible voters, is likely to be the largest exercise of democracy in human history, and even more likely, it seems, to be the object of immense indifference to the vast majority of people watching HBO at 11pm. The segment had some good lines. “Let’s deal with Gandhi first,” Oliver said to tee up a bit on Indian National Congress candidate Rahul Gandhi. “And I realize it’s not the first time that sentence has been said in a British accent.”

Oliver’s method is to ridicule the media mainstream for what it fails to cover and for its failures in covering the rest. His India segment on Sunday night did not find this kind of fresh approach. And his pleading at the end – “We should care about this story!”– didn’t help. There were other low points in the half-hour, including an outdated segment about truth in advertising, featuring a cancelled Kellogg’s breakfast cereal ad from “a few years ago”.

But then Oliver got his teeth into Alexander, the former NSA director, and all was saved. The value of the confrontation was not simply that Oliver made fun of Alexander to his face, although Oliver’s slyly abrasive handling of Alexander was a refreshing change from the glad-handing Alexander usually got in Congress. Oliver also showed himself to be an effective interviewer by coming back at Alexander with a quickness and precision that the former top spy has very rarely encountered in public.

Oliver and Alexander

When Alexander sought to justify the NSA’s bulk collection of Americans’ phone records by drawing the usual distinction between data and metadata, Oliver cut him off. “That’s not nothing,” he pointed out. “That’s significant information, otherwise you wouldn’t want it.”

“Is this the argument, then,” Oliver asked, “that to get the needle, you need the haystack?”

“Well, that’s part of the argument,” said Alexander.

“Right, but people’s concern, I think,” said Oliver, “is you’re not just taking the haystack, you’re taking the whole farm, and the county, and the state, and you’ve got some photos of the farmer’s wife in the shower as well.”

Alexander was to come in for still worse (watch the full interview here). The spot culminated with Oliver’s suggestions for ways to rebrand the NSA, because, “as we learned with Blackwater, you do not have to change the substance of anything you do, as long as you visibly rebrand.”

First suggestion up: the Washington Redskins. “It’s a slightly less tainted brand than yours,” Oliver observed.

“Yeah, but, probably not a good one to go with,” Alexander said.

The interview saved the week – and it worked because the host was out asking questions.

• This article was corrected on 28 April 2014 to state that India is holding parliamentary, not presidential, elections.