Their eve-of-poll headline, "Vote for Them", was aimed at the wives and sweethearts of servicemen still overseas. The subliminal message urged the women to put aside their scruples about socialism and support candidates who would be the choice of the men at the front.
From then on The Mirror occupied a special place in Labour's affection. It became "our paper", the one national daily on which the party could rely, and the Labour leadership's channel of communication to the working- class voters. In 1951 it cut back the Conservative majority by asking "Whose Finger on the Trigger?". Even in 1955, when a second Labour defeat seemed certain, it urged voters to "Keep The Tories Tame" - conceding victory but not changing allegiance.
So it is not surprising that John Prescott - standing up for what we coyly call "traditional values" - should have reminded New Labour that it had obligations to the citizens of Mirrorland as well as to the inhabitants of supposedly more prosperous Daily Mail country.
The comparison was a first-rate piece of political shorthand. But it was a gross over-simplification of newspaper sociology. The Mirror now has a working-class rival. The Sun is just as likely to be seen sticking out of building workers' back pockets. And the sons and daughters of the women who "voted for them" back in 1945 have, in many cases, acquired the social aspirations which The Daily Mail so successfully represents.
To succeed - indeed to survive - The Daily Mirror had to move with the changing class structure. In 1972, Hugh Cudlipp (the man who invented the "Vote for Them" headline) sent me to America to investigate and reveal the horrors of private medicine. The working title of the series was "Bleed Now, Pay Later". It is hard to imagine the modern Mirror - or any national daily - commissioning such a conscious exercise in class warfare.
The theme - confirmed by my experiences in Boston General Hospital - was that, once the well off began to invest in private healthcare, the quality of service available to the poor automatically deteriorated. No doubt Bupa prefers to advertise in The Daily Mail. But many Mirror readers will at least aspire to buying something better than is generally available.
The Mirror will not devote many column inches to arguing that - in the words of another of its ancient headlines - we are all "Members One of Another". The "red top" tabloids devote less and less space to serious political argument. They have been driven downmarket by The Sun. And the Labour Party has absolutely nothing to gain from its old ally behaving like The Daily Herald and boring itself into extinction. So Tony Blair should not expect The Mirror to convert many working-class Tories. But he must understand that it maintains an importance which is more than symbolic.
The Mirror represents - for it is read by - Labour's core vote. Thousands of its readers buy it every day because they have been brought up to believe that it supports the party. For the paper to maintain that allegiance is almost as important to The Mirror as it is to the Government. The problem arises - for both Prime Minister and editor - when the typical Mirror reader begins to feel that it is defending ministers who no longer represent their interests.
In the darkest days of Labour fortunes, The Mirror stayed loyal. That was not only because of the convictions of its owners and journalists. The Labour voter created The Mirror's niche in the market. Conversely - a truth that Tony Blair seems unable to understand - The Daily Mail and The Sun softened towards the Labour Party not because they were attracted by New Labour's policies but because their readers had grown tired of the Tories. Labour did not win because Rupert Murdoch gave the party his support. Mr Murdoch was suddenly converted to socialism when he knew that Labour was going to win.
Labour would be immensely damaged if The Mirror suddenly attacked the Government on behalf of readers who feel neglected by the party - not the aspirants to private medicine, but inner-city families at the end of long hospital queues. Commercial necessity will make the paper represent their interests. John Prescott might have said - and perhaps was saying - that the attitude of Mirror editorials might become a barometer of the disenchantment of Labour's traditional supporters.