There were two D-days in June 1944. The landings in Normandy on 6 June, Operation Overlord, recalled so movingly a fortnight ago, are part of British national memory. The other D-day remains virtually unknown both here and in America. Yet it was equally important in ending the second world war. And it also marked the dawn of cold war Europe.

On the night of 21-22 June 1944 the Red Army launched its summer offensive in Belorussia, three years to the day after Hitler invaded the Soviet Union. In 1941 the Germans had achieved total surprise, encircling millions of Russian troops and thrusting right up to Moscow and Leningrad. In 1944, however, the tables were turned. Operation Bagration, named for a Tsarist marshal who had fought Napoleon, hit the Wehrmacht with no warning. In five weeks, the Red Army advanced 450 miles, driving through Minsk to the outskirts of Warsaw and tearing the guts out of Hitler's Army Group Centre. Nearly 20 German divisions were totally destroyed and another 50 severely mauled – an even worse disaster than Stalingrad.

This stunning Soviet success occurred while Overlord was still stuck in the hedges and lanes of Normandy. Not until the end of July, as Bagration finally ran out of steam, did Eisenhower's armies break out and race across France to liberate Paris on 25 August and Brussels on 3 September. Overlord and Bagration together delivered the double whammy that knocked out the Thousand-Year Reich. At last, Nazi Germany was fighting a two-front war in northern Europe – the dread scenario Hitler had managed to avoid since 1939 – and the German people could now see the writing on the wall. It is no accident that on 20 July dissident officers tried to assassinate the Führer in a brave but quixotic bid to make peace before Germany was ruined.

Bagration helped to end the war, but it was also a sign of things to come. As the Red Army neared Warsaw, the Polish Home Army rose up against brutal Nazi occupation. Soviet forces were exhausted and in no position to fight their way into a major city, but Stalin's refusal to provide even token support for the Poles, or to let British and American supply planes use Soviet-controlled airfields, sent a chilling message to his western allies.

Much of Poland had been subsumed in the old Tsarist empire. In 1920 the Bolsheviks and the Poles fought a brutal war over the borders of newly independent Poland, which saw Polish troops briefly capture Kiev before being driven back to Warsaw. Two decades later, Stalin was determined to settle the question. In 1940 he secretly massacred much of Poland's officer corps at Katyn; four years later he happily watched the Germans crush the Warsaw rising – describing its anti-Soviet leaders as a "handful of power-seeking criminals" – before overrunning the country at his leisure.

In early September 1944, with Eisenhower's troops surging into the Low Countries, it seemed that the second world war might be over by Christmas. But then the Allies failed to cross the Rhine and the western front got bogged down. In British memory, the autumn of 1944 centres on that notorious "bridge too far" at Arnhem, but meanwhile, on the eastern front, Stalin made yet more dramatic breakthroughs as the Red Army smashed through Romania and Bulgaria into Yugoslavia and Hungary. The leader who, little more than a year before, had controlled only two-thirds of his own country now dominated much of eastern Europe.

During the cold war, the Yalta conference of February 1945 was often stigmatised in the west as the moment when Roosevelt and Churchill "handed over" half of Europe to Stalin. In reality, there was no handover in 1945 but a land grab in 1944, a byproduct of German defeat. By the time of Yalta, the Soviets controlled Poland and much of the Balkans: as Roosevelt admitted privately, all he and Churchill could hope to do was "ameliorate" that situation.

As important as Yalta was Churchill's meeting with Stalin four months earlier. Although an ardent foe of what he once called "the foul baboonery of Bolshevism", Churchill entertained a paradoxical faith in the essential decency of Stalin, born of two intense, boozy summit meetings in 1942 and 1943. The Soviet leader, though tough-talking, came over as unpretentious and businesslike, with a dry sense of humour. "If only I could dine with Stalin once a week," Churchill told a British journalist, "there would be no trouble at all. We get on like a house on fire."

In that spirit Churchill flew to Moscow in October 1944, seeking to agree on the shape of the postwar Balkans before the Red Army closed its grip. The result was the now notorious "percentages" deal concluded with Stalin late one night in the Kremlin. Churchill's aim was to preserve British influence in Greece and hopefully Yugoslavia. He did secure the former, often saying in later life that Stalin "never broke his word to me about Greece". But that was gained by effectively acquiescing in Soviet predominance across most of the Balkans.

By the time of the percentages deal, let alone Yalta, diplomacy could in fact make little difference. The new map of Europe had been decided not at the conference table but on the battlefield. And in that bloody story the other D-day of June 1944 should not be forgotten. "This war is not as in the past," Stalin told one Yugoslav communist: "whoever occupies a territory also imposes on it his own social system … as far as his army can reach. It cannot be otherwise." Soviet paranoia about security was understandable after the loss of 28 million citizens. But their obsession with a buffer zone in eastern Europe would define the cold war, at huge human cost. And the loss of that security blanket still haunts Putin's Russia.

• David Reynolds's most recent book is The Long Shadow: The Great War and the Twentieth Century.