Ed Miliband must prevent Labour returning to a "comfort zone" in which it attacks all government cuts without providing a compelling alternative for Britain, Tony Blair has warned.

Amid deep concerns among his supporters that Miliband mistakenly believes that Britain is moving to the left, the former prime minister says Labour must remain firmly rooted in the centre ground.

Labour issued a pointed response to Blair's intervention in the New Statesman, saying that Miliband is "challenging old ways of doing things", notably on immigration.

A spokesman said: "It is by challenging old ways of doing things, showing we have understood what we did right and wrong during our time in office that One Nation Labour will win back people's trust."

The rapid response came after Blair used an article for the centenary edition of the New Statesman to issue a warning that Labour must not become a protest party.

He wrote: "The guiding principle should be that we are the seekers after answers, not the repository for people's anger. In the first case, we have to be dispassionate even when the issues arouse great passion. In the second case, we are simple fellow-travellers in sympathy; we are not leaders. And in these times, above all, people want leadership."

Blair directly challenges Miliband's belief that the financial crash and the subsequent recession has created an appetite for a remodelling of capitalism from the left. The former prime minister writes: "The paradox of the financial crisis is that, despite being widely held to have been caused by under-regulated markets, it has not brought a decisive shift to the left. But what might happen is that the left believes such a shift has occurred and behaves accordingly."

Blair does not say Miliband believes this. But supporters of the former prime minister fear the Labour leader is in the process of making a mistake of historic proportions in believing that the financial crash, and a prolonged recession exacerbated by spending cuts, has created an appetite for a shift to the left. Some Blair supporters fear a repeat of 1987 when a reasonably united Labour party, which ran a slick election campaign, went down to a heavy defeat because it failed to tack to the centre.

Blair writes: "Having a modern vision … helps avoid the danger of tactical victories that lead to strategic defeats. It means, for example, that we don't tack right on immigration and Europe, and tack left on tax and spending. It keeps us out of our comfort zone but on a centre ground that is ultimately both more satisfying and more productive for party and country."

From left: Ed Miliband, Nick Clegg and David Cameron. Photograph: Reuters

He says the Tories face a "less menacing" scenario in which they are vilified for spending cuts. But they will say they are restoring order.

But Labour faces a more difficult challenge: "For Labour, the opposite is true. This scenario is more menacing than it seems. The ease with which it can settle back into its old territory of defending the status quo, allying itself, even anchoring itself, to the interests that will passionately and often justly oppose what the government is doing, is so apparently rewarding, that the exercise of political will lies not in going there, but in resisting the temptation to go there."

Blair says Labour will win permission to govern only if it provides answers to a series of policy challenges. He lists seven:

• What is driving the rise in housing benefit spending, and if it is the absence of housing, how do we build more?

• How do we improve the skill-set of those who are unemployed when the shortage of skills is the clearest barrier to employment?

• How do we take the health and education reforms of the last Labour government to a new level, given the huge improvement in results they brought about?

• What is the right balance between universal and means-tested help for pensioners?

• How do we use technology to cut costs and drive change in our education, health, crime and immigration systems?

• How do we focus on the really hard core of socially excluded families, separating them from those who are just temporarily down on their luck?

• What could the developments around DNA do to cut crime?

A Labour spokesman said: "It is always important to listen to Tony Blair because he has important points to make, including in this article where he emphasises our top priority must be growth and jobs.

"As he was the first to recognise, politics always has to move on to cope with new challenges and different circumstances. For example, on immigration, Labour is learning lessons about the mistakes in office and crafting an immigration policy that will make Britain's diversity work for all not just a few."