It was an unlikely setting for the prime minister to pay a compliment to the man who seeks to unseat him. But in closing the party conference season David Cameron left no doubt that Ed Miliband had succeeded, in the argot of elite sport, in getting inside his head.

A speech that brought praise from allies relieved to hear at last a coherent case for Cameron Conservatism was nevertheless strikingly defensive, as if drafted with the Labour leader hovering over the author's shoulder.

That much was obvious in the attempt to repel last week's raid by Miliband on the traditionally Tory terrain of one nation. "We don't preach about one nation but practise class war," Cameron said, aiming to shove Labour's tanks off the Conservatives' lawn. He later quipped that Miliband's monomaniacal insistence on stimulus spending meant that Labour was "the party of one notion: more borrowing".

Proof that Labour attacks have got under the PM's skin – or at least sufficiently worried his pollsters that they had to be parried – came in a high-risk passage in which he spelled out the criticisms one by one. "'Tory cuts, slashing the state … cruel Tories, leaving people to fend for themselves … elitist Tories, old-fashioned and out of touch.'" Of course he countered each charge as he made it, but one assumes an enterprising Labour activist has already chopped the clips together to make an online video: "David Cameron: in his own words."

There were subtler signs too that the Tories take Miliband much more seriously now than they did a fortnight ago. Among them: a section in Cameron's speech headed "Labour" that was longer than the one labelled "Economy"; a detailed response to Labour's demand for a Plan B, arguing that any new borrowing for investment would drive up interest rates; and disparaging references to "intellectuals of other parties" and "Labour party theorists" which surely pointed to one man in particular. Even an otherwise affecting passage on Cameron's father seemed designed to make a tacit contrast with Miliband's personal history: "Not a hard luck story, but a hard work story."

Glad though the Conservative tribe were to see their leader hit back at Labour, the source of the warm welcome they gave him lay elsewhere.

They applauded the speech because it dealt, methodically, with several of the anxieties they have had about Cameron and the government he leads. Plenty of Tories have complained in recent months about a sense of drift at the top, worrying that the government is buffeted by events – from Andrew Mitchell's pleb problem to the west coast rail fiasco – with no clear story to tell.

So they were relieved to hear Cameron in crisp, cogent form with a speech that, if it lacked rhetorical pyrotechnics, had a solid structure.

He came to make what he called a "serious argument". As Tony Blair had done, he used his conference platform to warn the country that in a world turned upside down by global competition, Britain risked falling behind if it did not sharpen up its act. We could either be "lean, fit" and on the rise – or else join those nations who were "fat, sclerotic" and on the slide.

To secure our place in the former category, we would have to sort out our deficit, welfare state and education system. Once we had, we would see a country of social mobility in which those striving for success could "achieve [their] dreams".

In making that case, Cameron conveniently focused on the three areas where the coalition is united and Labour is weak: polls show Miliband lags on the economy while his party has been hesitant on education and welfare, unsure whether to oppose government reforms or back them as completing a job Blair started.

But Cameron's core argument also dealt with several of his internal critics' nagging concerns. First is the fear that, by focusing on the deficit, the coalition looks like a government of bean-counters, obsessed with the price of everything, appreciating the value of nothing. So the PM insisted that deficit cutting was not merely an economic mission but "also a moral one", because only in a sound economy could people fulfil their potential.

This was lapped up by the ConservativeHome wing of the party, which has long urged Cameron to embrace a muscular brand of compassionate conservatism, acting for what it calls the "strivers".

The PM's "aspiration nation" slogan may not take off with the public, but it will placate those Tories who have feared that Cameron is fast looking like the head of a government by toffs for toffs. He gave them soundbites to savour, insisting that the Conservatives were the "party of the want to be- better off" and "that I'm not here to defend privilege, I'm here to spread it".

All of this amounted to an answer to a question Tories have long wanted Cameron to address. Given that he can't say how long Britain will have to keep taking the austerity medicine, they have yearned for him to explain again – and more effectively – why the treatment is necessary and what the country will look like once it's over. On Wednesday he made a decent stab at that. The result is a conference season that leaves the three main leaders stronger than they were before, at least with their own parties, and which has opened up a genuine and substantive argument between the main two. Both Labour and the Conservatives are now locked in a fight for the centre ground, each claiming to be the authentic voice of One Nation. The shape of the next general election just got clearer.